If the American colonies stayed in the British Empire it would have to peaceful. I think that the Parliament would be able to get the Americans to pay taxes, but they had to wait for an Empire-wide crisis. I think that crisis would be extended war- which I will explain below.
IMO between 1763 and the (IMO inevitable) French Revolution (which HAS to start in '89 because of the crisis in food prices, the factor which also sparked the '30 and '48 Revolutions) there is going to be another Anglo-French War. So lets say a healthier Pitt the Elder, who holds off American taxation, and a war that starts in the 1780's between the French-Austrians and Anglo-Prussians sparked by a Dutch Civil War between the Organists and Patriots.
So the Continental throw-down occurs in the 1780s, probably during the new ministry of William Pitt the Younger. The expense of the war (the French will be spending even more money than they did during the ARW- since the Austrians were basically unable to fight a war w/o a foreign subsidy) drags down French finances. France, tied to an incompetent Austria and facing a determined Britain, loses the war.
The war ends in the late 1780s with the Second Treaty of Paris (say 1787). In 1789 the Estates-General are called, and we have the French Revolution. IMO the American Revolution was not nearly as important an intellectual factor in bringing on the French Revolution (obviously the financial stress of the ARW was a major factor) as the Enlightenment thinking that intellectually drove both Revolutions.
The French Revolution rapidly gets violent, and they execute Louis XVI in 1791, forcing a general European war.
I think that if you can keep the Americans in the Empire until the French Revolutionary Wars, then you've permanently kept the Americans inside the Empire. The Revolutionary Wars are going to allow lots of pro-British feeling, with the kind of territorial gains in the Americas that will attract popular support among Americans (Louisiana at a minimum, perhaps also Cuba?). The British had planned on launching a major war in South America in support of South American independence leaders (Wellington was originally headed to Venezuala before the French invaded Spain). So American troops are fighting and dying for the Empire, alongside troops and officers from the UK.
The whole key is for Parliament to come up with a way so that the Americans share the financial burden of the Empire, while avoiding direct taxation of the Americans. Some kind of reorganization of the American colonies, so that they can be made responsible for their own defense might do it, shifiting the financial burden onto the Ameicans. Delegating oversight of settler-Native American relations to this new body would also be important, so that the Americans can fight Indian Wars on their own.
Under a longer Pitt the Elder ministry, the Albany Plan is adopted, with Benjamin Franklin selected to be the first Governor-General (or whatever its called), would be a total cop-out, but I think that if Mr. Franklin's new colonial government is given the power over Indian relations and the land west of the Appalachians then it could have enough heft to both unite the colonial governments and allow Parliament to shift financial burdens to that body. If Parliament also assinged the Albany body oversight of enforcing the Navigation Acts, that could work too.
To answer the POD, I think that a British North America would be just as expansionist as its OTL counterpart. Moreover, I think that British North America can be just as expasionist as OTL, while the rest of the British Empire proceeds a pace. This combined Anglo-American Empire is going to be overwhelmingly powerful by the mid-19th century. I think that the 20th century in this TL is 'Pax Brittania' with the Anglo-American Empire keeping the peace.