Well, thinking of a BA in the UK as 3/4 of a US BA seems very naive to me, to be honest. It's a completely different system of education. In the US, your first year is spent on studying many things not related to your major, whereas your BA in the UK focuses on your subject right from the beginning. I don't see how research in the UK is less "real" than research in the US, maybe you would care to explain.
Anyway, I think you're right about arranging some research with a professor. Not sure how easy this is, though. I don't know if professors are willing to accept students from outside their own university to conduct research with them.
Yes, I'm familiar with those details of the U.K. system, but even if you spend your first year in US "studying many things not related to your major" you probably still do some core major requirements at a college level of rigor.
But mostly I'm just repeating what I've heard; if that perception is held by others in the schools you applied to, it might help explain your failure to get in their graduate programs (then again, the acceptance rate at the top schools is very low. At MIT it's like 12%? 2.5%? I forget exactly, but I think more like the latter of those two figures).
The comment about real research wasn't linked with anything having to do with the U.K. per se, just that you now have an serious opportunity to do "real" research undistracted by school.
I'd approach professors not as a student pe se but more as a potential research associate (e.g. full time lab worker) who's happy to do coding and the like for free. Make it clear that you're establishing a foundation for your post-graduate study, but focus on what you can do for them.
Anyway, I think you're right about arranging some research with a professor. Not sure how easy this is, though. I don't know if professors are willing to accept students from outside their own university to conduct research with them.