Can I negotiate a research assistant position if my PI moves to a new univerisity?
I think it is more than reasonable to ask for a "bump", but you may want to reconsider moving. The way I see it is you are currently in your 2nd year as a post doc and the move isn't for 9 months so at the time of the move you would be approaching 3 years as a post doc in the same lab. This is getting to be a long time and you do not want to lose the time associated with the lab move to move to a new lab for a year. You need to make the move to the new lab desirable enough that you want to stay for 2+ years. The reason you may want to reconsider the move is that you would be effectively doing 2 post docs in the same lab instead of 2 labs. This means less of a chance to learn new things (both technical and interpersonal skills) and meet new colleagues (specifically ones who can write letters of reference for you and help you get the next job).

If you chose to stay, the argument for a bump is that (1) this would effectively be your second post doc, and (2) you would be taking a hit in your research associated with the move and need additional support to over come that. The reason keeping you is advantageous to your PI is continuity and the ability to get started quicker. Finally there is the issue that the PI probably has a start up package so can actually afford a bump (both salary and resources).