To simply address this, no, bounties were never placed on any of the Founders’ heads during the War of 1812.

To fully understand the answer, it needs to be understood that by 1810s, the British weren’t harboring 30+ year resentments against the Americans for their rebellion. Funding and supporting the American Revolution was not unanimously popular in Parliament in the 1770s, and by 1781, the year of the Battle of Yorktown, major factions within the British government wanted the war to end, even if that meant losing the colonies. The American Revolution was not a popular war in Britain in the 1770s and several factors made it continue to lose popularity as the war continued.

What should be stressed as well is that the American colonies at that time were not particularly valuable when compared to the other colonies spread around the world. Britain’s economy was bolstered by their colonies in the Caribbean and India, rather than in the 13 colonies. This doesn’t mean that Britain didn’t care about the US, but after France joined the war, which effectively turned the war from a regional war to a world war, Britains didn’t strive to keep the colonies at all costs. And it is worth noting that shortly after the Treaty of Paris, which ended the war, the British began more normal relations with the United States (founder, John Adams, even personally met with King George III in June 1785), so by the 1810s, Britains were less concerned with the loss they had in the Revolutionary War, and were more concerned with the current conflicts in Europe and North America.

On a similar note, the British were not in the habit of making bounty lists against enemy leadership. I answered a similar question if there was a most wanted list by the British against the American founders during the Revolution, which even during the war, didn’t happen.