I'd just like to add to the list of Americans with American degrees who got library jobs in the UK. My husband's first job here was as an Information Assistant in a university library. Not a librarian job, I know, but if he had wanted to pursue that career it would have been a good start. He has US degrees in history and politics.
HOWEVER, that was while he was on the International Graduates Scheme visa, so no need for sponsorship. I agree with the other contributors that it would be extremely unlikely to get sponsored for a librarian post as a non-EU citizen without an existing right to work in the UK.
Having said that, I completely understand your frustration, LaraMascara, especially with trying to convince employers that you really do want the job you're vastly overqualified for! Sometimes it can seem like it really is an anti-American thing, especially when you're feeling demoralised from the job hunt. During my husband's job hunts over the years we have found that ignorance of visa rules on the part of HR departments was a constant pain, and we were always happy when the initial application form didn't ask any nationality/visa questions. Also, he once walked into an interview when the first words out of the interviewer's mouth were "Oh, you're American..." Followed by an awkward silence. Not sure what was going on in their mind, but he wasn't offered the job.