If the American Revolutionary War was put down militarily how would this impact British Politics? Would the military victory manage to keep the Tories popular enough to retain the government? For the sake of PoD let's say Washington is killed at the Battle of Brandywine and the war is over by 1780.
Yes. George III heavily favored them, the only reason they lost power was the war.
To answer the question I'd say British politics swings quite a bit more conservative, likely putting politicians like Pitt the Younger and Edmund Burke solidly on the Whig side. Lord North or members of his cabinet probably remain in power for years longer, he was George III's most favored and most effective politician.
Eventually there's going to be some sort of crunch time as the British realize the American colonies are expensive to keep under control and dangerous in the event of a European war, and meanwhile calls for parliamentary reform and the rise of radicalism are going to continue to get louder and louder in Britain, so within a few decades of the victory I would expect a major shakeup in both colonial policy and domestic policy. I'm guessing this would
not come fast enough to prevent another revolt, but you never know. Lord North and his ilk are not going to want to change their policy towards the 13 colonies, and I would expect them to try and hang on to a relatively hardliner stance for another couple decades until they inevitably lose power. Given that OTL the US population surpassed that of England by 1840, that's a pretty narrow window for a second revolt to not happen.
And of course another big question mark is how this affects the French Revolution.