Interesting,

Tell them you have paper trail that prove the prof lied to you about funding. If they give you the package, you might not sue, if they don't, you definitely will sue. At least you will go public with all the information you have. It might be hard to make a case in the court of law, but it's much easier to throw mud.

Just bluff them first. Don't put in writing. See what happens. You have time to play anyway. But don't put too much hope in it.