Bah. Owing to my Sans-Cullote hair, no doubt, I feel people tend to misunderstand the French Revolution.
There's no doubt that the American one was, in some very important ways, more succesful. It suceeded in its goals with minimal upheaval and ended with establishing a rational, flexible, durable, intelligent constitution who's ideals remain relevant today.
It also left thousands and thousands of people literally enslaved. Sure, trying to change that could well have created Haiti times ten to the twelfth; practically speaking, it was probably the better option. In terms of principles, it's still abhorent.
I broadly agree that if the French revolution had stopped where the American one did, it would have been better for everyone. Better for a lot of the people killed by the wars, not least Frenchmen, and better for future generations lacking the meme that you can just unplug a society and start again. But...
You have to realise just how miserable, wretched, and inescapable the conditions of the vast majority of people in 18th century Europe - in France, Britain, anywhere - actually were. Most people lived under the rule of a hereditary despot kept in power by a mixture of fear and tradition who could waste his wealth on stupid prestige projects and military willy-waving while his people eked an existance of hunger, overwork, and brutalisation. "North Korea", in other words. Sometimes, it was somewhere under 10% of the population rather than just one guy. "Apartheid".
The usual assumption is that the moment that dictators and executions came along, the Spirit of the Revolution, generally imagined to be that of modern liberal democracy, was betrayed, and it was all going wrong. But to think in contemporary terms, people really were willing to go through a lot of hardship, and to accept the necessity of obeying orders and killing lots of people, if they thought it would overthrow the system that governed lives they'd been born into - and given those lives, that's pretty understandable.
In short, it's easy and probably correct to be a Girondist now, but it was easy to be a Jacobin then, and I think many of us probably would have been.
The point of all this is that making the revolution less radical (which almost certainly means keeping the king) is unlikely to inspire a revolution in Britain. A moderate revolution would be a mechanism for the wealthy middle classes to gain a political say and a fairer system of finance, which in Britain they already had the beginnings of, comparatively speaking. And on the other hand, I don't think a radical revolution is at all impossible in Britain.
One way to go about it might be to have us win early on in America. The precedents this established - that if a legislature is troublesome you can shut it down, and that people outside the established church are hiding under the bed working against the King - combined with the prestige it gives to the hardliners could swing Britain further into reactionary authoritarianism than we already went.
France is still going to be the spark, because France is the one with no dosh (the losses of the ARW can likely be made somewhere else, even if it delays things a little). But once that happens... well, in the latter 1790s, things got pretty hairy for the government. Add in a messy situation in the colonies, and you could have a triple-whammy.