I'm surprised nobody has pointed this out, but you seem to be confused about the NIH Public Access Policy. It does not require that authors keep their copyright (journals would not have accepted such a policy, I'm sure), but that they make their article freely available on PubMed within 12 months of publication. There has been discussion of NSF having a similar policy (which would be great), but this would mean little change in the situation for mathematicians, as it would just be a requirement for people to put their work on the arXiv, which anyone with any sense does anyways.
I honestly would be very curious what would happen if you tried to negotiate with the journal on the copyright issue, but I'd doubt you'd get far. (EDIT: Of course, as was noted above, some journals don't require authors to transfer their copyright, in which case things are fine.) At the end of the day, this just absolutely the wrong point on which to pressure a journal (and in my opinion, not a very important criterion when choosing what journal to publish in). They don't need your article; no offense, but I doubt replacing your article with the next one they would rejected is going to cause publishers any worry. Like Matthew, I think approaching the problem from this angle as a graduate student could only be cutting off your nose to spite your face; the opportunities for damaging your career completely outweigh any societal gain. I'll admit that the idea handing the copyright for a paper to a journal rankles a bit, but it certainly has yet to make the least difference in my life, and I don't anticipate a time when it would. Anyone who wants my papers can get from the arXiv, and they'll continue to, at least until Skynet becomes self-aware.
The papers it's very annoying that publishers own the copyrights to are the old ones, since that actually does limit access to them, but that milk is already spilt.