I'm American and I've heard Americans say "I beg your pardon", but like you I've always thought of it as a slightly proper (maybe WASP-y) idiom. People frequently say "excuse me", "sorry?", or "say again?". At least I do. Maybe I should get my ears checked.
I'm familiar with the expression but if an American said that to me, I'd probably think it meant "rethink what you just said".
It depends a lot on their tone. Most of the time I've heard it, it's a quick "begpardon?", sometimes with their ear cocked towards you.
When I see it in writing, I too for some reason picture an angry posh British man who is about to demand satisfaction.
They'll also commonly say "pardon me", which is a bit nicer "say again", but definitely nowhere close to "I beg your pardon" uptightness.