Should I offer to pay (some of) the travel cost from the UK to the US for an interview?
Would the potentially high cost of flying me to an interview lead to some institutions passing me over?


I believe many universities, especially smaller universities, will balk at flying overseas candidates in for interviews. Often they will move down their short lists and only if they are unable to find a suitable candidate that is higher than you on the list, will they be willing to fly you in. This isn't a huge disadvantage, but it is a disadvantage. In my experience candidates often hurt themselves during interviews and rarely perform so well they substantially move up in the rankings.


  If so, would it help if I offer in the cover letter to (at least partially) pay my way?


I would not offer to pay for an interview directly. This would in essence be calling out the search committee for being cheap. A better strategy would be to mention in your cover letter that you will happen to be in the States, possibly even states/cities near by, on a couple of dates and you would be happy to extend your trip and come to them for a visit. It would be nice if you have an academic pretence for those visits (conference, seminar, or visiting colleagues), but even saying you will be in the States for personal reasons is fine. If you do not get an interview, you do not have to go to the States.