No, I don't think so.

In the 1770s, England had a population three or four times greater than the British North American colonies, but demographic trends made it clear that the Americans would one day outnumber the English. In a publication in 1755 (Observations Concerning the Increase of Mankind), Benjamin Franklin had concluded that the population of the colonies was doubling every twenty-five years. If this was correct, as it proved to be, the population in America would exceed the population of England by the middle of the 1800s.

Given these demographics, what would representation look like in Parliament?

First, during the Revolutionary period, there were some 550 members in the House of Commons. Representation, even with England, was very uneven. Old towns that had declined long ago and had tiny populations might still have the right to send men to Parliament, while new and flourishing cities (like Manchester and Birmingham) sent nobody. And yet, the British still maintained that Parliament had authority over the whole -- the idea that "people could be represented only by men chosen by them from their geographical area" did not match how Parliament actually operated.

The Americans, however, were increasingly arguing that government should work that way. But even if the British had accepted that idea, how would it have worked? Would Parliament have been willing to assign one-fourth or more of their seats to the Americans? Would they have been willing to increase that number as the American population grew? No, the very idea of handing so much power to the colonists would have been unthinkable. Or, would the Americans have been content with 1 delegate, or perhaps 13? Not a chance. They would have recognized that such a token membership would give them no real power, while simultaneously undercutting their argument about "no taxation without representation."

The Americans and the British were increasingly making different arguments about what representation meant and who actually represented people living in the various parts of the Empire.

1. The British said that Parliament had legitimate authority over all the empire, whether people were living in England, or Scotland, or America, or India.
2. The Americans increasingly argued that Parliament could only represent the English. It could not represent the Americans, because Americans did not vote for members of Parliament. Who represented the Americans? Why, the elected members of their own colonial assemblies, of course!

This is two very different models of Empire. The British model suggested that King and Parliament (both located in England, of course) ruled a global empire. The American model proposed that all parts of the Empire shared a common loyalty to the king, but that each part of the empire had (or should have) its own representative legislature. No legislature could make laws for people in a different geographical area. So, Parliament truly represented the English and had authority over them. But it did not have authority over Virginians. Their own elected body (the House of Burgesses) had legitimate power there.

The English regarded this argument as ridiculous. The idea that a backwater government in Georgia or Rhode Island could claim to be equal in dignity to Parliament struck them as bizarre. The idea that the legislative body of Great Britain had no power in the British colonies was equally ridiculous. After all, hadn't the colonies developed under British authority? Didn't they benefit from British trade and British military protection? To put it in modern American terms: to British ears, this sounded as if a high school student council was claiming to be equal in dignity and authority to the U.S. Senate.

In 1766, Benjamin Franklin was questioned by the House of Commons about the Stamp Act. He said that that Americans did not see themselves as being represented in Parliament, and therefore would never pay the tax. He proposed a solution, however: rather than trying to tax the colonists directly, Parliament should instead simply ask the various colonial assemblies to tax their own citizens. He assured his listeners that the Americans would generously show their support, and the taxation problem would be solved. Parliament, of course, did not listen to this proposal.

If Parliament had listened, would it have worked? Would the American assemblies have taxed themselves as Franklin assured the House of Commons that they would? Could we have avoided the Revolution that way? Well, one can never be sure. Anything is possible. But realistically? No.

How do I know? Because when the Americans had the chance to put their own government together, under the Articles of Confederation, they worried a great deal about centralized power and the destructive power to tax. So they gave Congress no power to tax at all, and said that when Congress needed money, it could simply ask the states for help. To put it mildly, this did not work. It was one of the primary failings of the Articles that led many of the nation's leaders to believe that a stronger national government was necessary. As James Madison put it, while arguing that a stronger Constitution (granting Congress the power to tax) was needed:

There is little reason to depend for necessary supplies on a body [any of the states] which is fully possessed of the power of withholding them. If a government depends on other governments for its revenues; if it must depend on the voluntary contributions of its members, its existence must be precarious. A government which relies on thirteen independent sovereignties, for the means of its existence, is a solecism in theory, and a mere nullity in practice.

Sources:

Benjamin Franklin Before the House of Commons, at https://founders.archives.gov/documents/Franklin/01-13-02-0035

Soame Jenyns (mocking the idea of "no taxation without representation") at http://www.let.rug.nl/usa/documents/1751-1775/soame-jenyns-the-objections-to-the-taxation-considerd-1765.php

James Madison, "Weaknesses of the Confederation" at https://founders.archives.gov/documents/Madison/01-11-02-0065

Without looking at the details, it does sound like a compromise that could have worked. Other responders have listed why it wouldn't have been adopted. I'll try to explain how, even if it had been adopted, it wouldn't have worked logisically. A couple bullet points:

• Too much time lag to be effective. There was a very long time lag present in all crises/reactions. Estimates on how long it would take to sail from America to England one-way ranged from a month to six weeks, and was never guaranteed to be successful at all. Imagine a crisis in America. It would take weeks for news of that crisis to reach the Americans who served as a conduit to Parliament, more time to get a messenger to the coast, maybe six weeks to sail to London, more time to get the crisis through the various stages of debate in Parliament and rubber-stamped by the monarch, and another six weeks to get the message back to America. Much longer than that if the response required military action. So a stimulus-response loop could have taken multiple months, which wasn't feasible when local initiative was needed (for example, if the crisis were "the French are invading from Canada.")
• Not enough representation to matter. Representation in Parliament would have been tricky because the overarching aim would have been to provide the Colonies with enough representation to keep them quiet, but too little to effect change that could affect England. This would have been pretty transparent to any American representatives serving in London, and that sense of futility would have fed the same frustration that no representation did. The # of constituents per member of Parliament would have been a smaller number for an England-based representative, a much larger # for a colonies-based representative. England's representatives could vote down the colonists whether that was in the normal bicameral structure or if a third house was added - a "House of the Empire" could get voted down by Lords or Commons. So the cry would have gone from "no taxation without representation" to "fair representation." Regardless, there was a line in the sand that England would have drawn regarding Parliamentary representation, and their refusal to change would be enough to create the same anger and frustration from the colonies that "no representation" had in the first place.
• Representatives would be out of touch with their constituents. This legislature would have been out of touch with colonists regardless of if the legislature included colonists or not. English representatives would be debating and making decisions on the colonies having never set foot there. Colonial representatives would be debating and making decisions on the colonies from an increasingly-inaccurate viewpoint (having spent increasingly more time in London and less time in Pennsylvania).
• Colonial representatives would be targets when they traveled. There would be some serious security issues present in this arrangement. Whenever colonial representatives go back to their constituent areas, they would be targets while in transit on the ocean. A French attack in the Atlantic could net them a handful of Members of Parliament and all the knowledge from the sessions of Parliament they attended, as well as the propaganda appeal of having captured representatives of your enemy. The English would need to sacrifice assets in those hostage negotiations, or risk enraging the colonies by refusing to negotiate.
• Everyone counts, or you decide who to leave out. This would set a dangerous precedent for colonial representation. If America gets representatives, does Australia and New Zealand? India? Do you draw a line, and where is it if you do? Whoever ends up on the wrong side will inherit that grievance, so we'd essentially be outsourcing "no taxation without representation" to whichever colonies didn't make the cut - if we ever let go of that grievance ourselves. If all colonies are added to Parliament, the logistical issues mentioned in the first bullet point would increase exponentially.

As I already stated, others have detailed very well England's stance on the issue. I was merely trying to explain the other side - "taxation WITH representation" would not have solved colonial grievances because of the way it would have been implemented.

In addition to the answers recieved you may enjoy this lively discussion based on essentially the same question answered by u/Jordan42 and some great follow-up answers by u/lobstahpotts (and myself).