Your challenge is to have George Washington and King George III meet at some point in their lives.
Not to hard really. Just have Washington, a decorated military hero, visit London after the French and Indian wars. I'm sure that the King will receive him at some public function, like a ball or a Drawing room. But it wouldn't have any real effect.
Now an interesting idea would be to have a friendship develop between Ben Franklin and George III. Perhaps the two could find a way to create a compromise between the Colonies and Great Britain.