I think you run a risk of losing the offer

Whether an employer will keep negotiating with you is not a rational decision, but an emotional one. Once they don't like you, that's it. They'll take the offer back. It is very difficult for an employer to set the precedence that they pick up the GC cost of your previous employer. That's why they just give you a sign-on bonus to make the purpose vague. When you are still in the GC process, you are actually very vulnerable. So be careful, you may not want to push your luck.