We'll Meet Again
A simple cut a paste with signatures on.
It is all down to humility and power sharing. Agitators wouldn't have much to work with without George III being such a ninny. Different succession? Either Frederick lives to succeed or George III abdicates (a letter was written in OTL, but not sent), although this is because of the war and a little late in the TL to help much. His successor (post 12 August 1762 so someone* would have to act as Regent for young George IV) would have France and Spain to deal with too. Although, if successful, he might obviate the Napoleonic Wars.
Prince William, Duke of Cumberland 'The Butcher' would certainly want to have a crack at it if his father had found a way to ensure George III was passed over as he wished. 'Accident'? A strong supporter of Georgia and inspiration for Handel's 'Hail the Conquering Hero'. He dies in 1765 so this would have to change or he would have to produce an heir.
*Prince William Henry, Duke of Gloucester? (Father to 'Silly Billy')
Perhaps a stronger Pitt the Elder (free of gout) could have imposed a general law of principle against taxing the colonies, yet his own determination to pursue war against France and Spain for a further year (Eight years war) would have made the need to pay down debt even more pressing. In hindsight a final victory over the French that puts an end to their naval ambitions would have saved far more money later (Corsica not annexed in 1768 so Napoleon isn't French, France can only support American rebels weakly).
A 1775 Conciliatory Proposition that allows colonies to remain tax free that meet their own expenses (cf that tax themselves). Although how that would effect enforcement of the Proclamation Line, I don't know.
How about no Seven Years War at all? Or not supporting Frederick the Great to the tune of £700,000 a year (1760s sterling).