It is an issue of rhoticity. Literally the only difference is the rhotic R. I say horse like sauce because I don't rhoticise the R. This doesn't make my horse sound like an American sauce - and why would it? Why would a non-rhotic speaker pronounce a word without an R anything like a rhotic speaker's R?

A non rhotic r in horse does not make a non rhotic r in sauce. That's not a question of rhoticity because how you pronounce the r sound doesn't matter....its that there's an r sound at all in sauce.

You agreed with this in another comment regarding the British pronunciation of sauce sounding like 'source'. That again has nothing to do with the rhoticity of the r in source, only that there is an r in sauce.

Yet here you refuse to come to the same conclusion that you did on another comment because 🤷

I am not saying this is specific to you, I'm saying this is a difference in the pronunciation of the word between british and american english. I think the issue here is the comparison to another word rather than someone just linking side by side pronunciations of the word in question: sauce. Horse and source are irrelevant. Side by side there is a clear addition of an r sound in sauce from American English to British. Neither is wrong or right, and there's nothing you should be getting offended over here.