Did any British PMs support appeasing the colonial unrest?
Well, "representation" had a much different meaning from what it has today, for most of Parliament, certainly for King George and peers like Lord North.
To quote Edmund Burke, Parliament was not " a congress of ambassadors from different and hostile interests, which interests each must maintain, as an agent and advocate, against other agents and advocates, but a deliberative assembly of one nation, with one interest, that of the whole, where, not local purposes, not local prejudices ought to guide, but the general good, resulting from the general reason of the whole". To the British parliament, therefore, the colonists had representation. As conservative pamphleteer Thomas Whately pointed out, plenty of people in England were, like the colonists, unable to vote- Birmingham and Manchester were very large but sent no representatives to Parliament. Nor did anyone lacking land get to vote. This is what Gordon S. Wood has called "virtual representation". This paternal attitude is of course very old, like the idea of a king looking out for his people's interests. It presumes that everyone has the exact same interest. For the colonists and England that was not the case. You can see the same sort of attitude much, much later, when women were asking to vote and men were saying, why, we work in their behalf already, why should they?
We think of elected representatives now as representing the wishes of the people in their districts. That would be a radical notion in 1775, something you would hear from people like John Wilkes, but from very few holding positions of power. The colonists, if they had actually demanded representation in the sense we think of it now, would have been asking to reform the basic structure of the English government. They did not even try. That reform would only happen in the next century.And when the Chartists first began to ask for it, they were fiercely repressed.
But as to whether there were members of Parliament who favored conciliating the colonists: definitely. One of them, Isaac Barré, did even more than speak in their favor. He had been in the French and Indian War, knew something about the colonists, and warned that the colonists were not going to respond well to suppression. He said it would make them fight, not make them back down. He turned out to be quite correct. When the Howe brothers sailed across the Atlantic to take charge of the situation, they came with concessions to give to the colonists- but because they were told to first teach the colonists a lesson with military action, things quickly escalated beyond concessions.