The road to war was not as well paved with Palmerston as you imply.

At the outset of the Civil War, British subjects invested a lot of their savings in American securities, around 27% of the exports of Great Britain went to the United States, and the British imported more than half their foodstuffs from the U.S. But none of that was more important than cotton, at least for a time. All of the other things I mentioned favored the North and the West, not the South. But cotton supported the basis of Great Britain's empire, the South's cotton represented three-quarters of the British market for raw cotton. Raw cotton was the commodity supporting the base of the industrial and commercial nature of the empire, and directly employed four to five million people. If the cotton stopped going to Great Britain, the economy may collapse. Palmerston knew that and saw supply of raw cotton to Britain as a vital British interest. However, the government was not entirely of one mind. Lord Lyons, the British minister to the United States was quoted as saying that the South had exaggerated and false ideas about cotton and that it could cause them to act stupidly.

The Confederacy was offering the European powers advantages based on the cotton trade, influence, and trade with the new nation in exchange for diplomatic recognition. The Union knew what the Confederacy was up to and Sewart sent a mission to the British court. When it arrived they learned that London had recognized that the Confederacy was a belligerent in the Civil War. That this wasn't a regular rebel group. The man responsible for dealing with the American missions in an official capacity was Lord John Russel. What greeted the Union mission was somewhat of a stalemate. the British government had formally stopped subjects from providing war materiel or aid to either side, which pleased the North since it could provide itself with most of what it needed. Seward, the American secretary of state had intended to send a threatening message with it's mission, threatening war if the British recognized the Confederacy as an independent nation but Lincoln himself stepped in and made the message a tad bit less belligerent. The message was never used because by that time, the North had lost at Manassas.

Even if the Union couldn't threaten it's way into winning, the Confederacy hadn't been able to make any political headway. A change was required. The Confederacy dispatched two additional diplomats to London, they were to reach London by sailing on the H.M.S. Trent in Havana. As the Trent left Cuban waters in November 1861, it was intercepted, the Americans captured the two new diplomats. When the British learned of their seizure, it set London abuzz; the stock market fell, Parliament was called in for a emergency session, and some newspapers started calling for war all the while insulting Lincoln for being so shortsighted. Palmerston instructed Lyons to demand the release of the diplomats, and to formally apologize. However, Prince Albert stepped in... He saved Anglo-American relations by changing the expectations of the government. He convinced Russell, Lyon's boss, to let the American's get away with the Trent affaire by claiming that the U.S. ship that seized the Trent hadn't acted under official orders, and not to insist too much on the apology.

Before the message reached Washington, the British government had gone cold on war because Canada had been understood to be rather vulnerable, they feared that the British navy wouldn't be able to crush the American navy fast enough to protect British shipping, and Prince Albert had suddenly died. Russell advised that war was not in the nations interest. He is quoted as saying that he did not think that the country would approve of an immediate declaration of war, so Lyons made the "ultimatum" over the Trent affaire something of a formality. The Americans received the ultimatum and Lincoln saw that there would be no way to fight two wars at the same time. Ultimately the U.S. and Great Britain had worked to defuse the crisis, the Confederate diplomats were released, and Great Britain had turned the heat down on it's message.

The Trent Affaire didn't grow into a war because both sides realized war was not in their interest. Palmerston was not pro-Confederacy, he wanted to weaken the United States and strengthen Great Britain, and he believed that war with the Union wasn't the way to do that, especially in light of the Confederate successes at the beginning of the war. The intervention of Prince Albert also proved to have had a huge impact, it gave the British government an excuse for being less belligerent. His sudden death also meant that they had an excuse not to go to war.

Sources:

The Civil War by Ward and Burns.

Battle Cry of Freedom by McPherson

The circumstances of the Don Pacifico affair were radically different to those of the American Civil War.