This is so wildly institution dependent that none of us can say what you can or cannot negotiate. When I was hired for my current job I was a full professor being offered an assistant position, which the provost bumped to associate without my even asking! So make some discreet inquiries, it might be possible and it might be absolutely impossible. Do make sure you are not missing out on a salary bump.
You were willing to go from full prof to assistant prof? Is that commonly done? Wouldn't that big a demotion look bad on one's CV? At my previous institution, the titles (full or associate prof) for incoming hires were always kept the same even if tenure didn't transfer, and I just assumed that was the norm.
I'm not LarryC, but your reaction prompts me to chuckle a bit.
I know of a couple of people who went from tenured full professor with international accolades to postdoc because that "demotion" fit better with their current career goals*. At some point, caring about the title is silly when one sees a position that one wants, the pay is acceptable, and practically nobody is ever going to look at the CV (would you check the CV of someone like Stephen Hawking?).
When very little is on your CV, then, yeah, titles matter. If, however, someone has to wade through several pages of publications and presentations to get to the list of positions held after years of interacting at conferences and reading many of those publications in the course of doing research in that area, then the probability that any one line is a deal breaker is low.
*In one case, the guy was a giant in the field, emeritus for a couple of decades, and just wanted a lab to visit a couple days a week when his primary institution kicked him out of the space he had occupied for decades for someone who was going to work 80 hour weeks and supervise dozens of grad students. No one thought less of him for officially being a "postdoc" in someone else's excellent lab.
