Responsibilities of PI to PhD student post PhD submission, but pre-viva
This answer is a bit blind as some information needed to give a more complete answer is missing. I'll try to give some guidance, but it is probably incomplete. 

You say that you haven't gotten along well with your advisor. You don't say why. There could be a lot of reasons, but if you have, through your own actions, given the advisor reason to question your commitment or ability to finish, then you may not get very far here. It is impossible to say what is "owed" in such a situation. 

You say you want to "fight for this project...", but I'd suggest that you change that approach a bit and stress why and how you are the best person to complete the work and how you guarantee that you can and will do it. Funding is a separate issue and, again, it is impossible to say what is "owed" here. 

Depending on the nature of the work, it may be completely reasonable for the professor to pass it on to someone else since you are no longer involved. In a lab situation, students very seldom "own" the research thread and that thread may be so important to complete that personalities matter less than the work itself - childhood cancer, say. 

Technically, your advisor owes you what was in the contract. Morally it may be more, but there isn't really enough information to judge it externally. 

On the other hand, an attempt to deny you your degree or prevent you from publishing your results or impeding your career would be unjust. 

If you want to stay with the project, however, convince your advisor that you can and will be the best person to do it. Any other approach is likely to fail.