What you did verbally, without documentation and contracts, is of little realistic value (other than to maybe help verify the fact that it looks like you're getting screwed).
From what you have written above, it looks like your company A can't do any business with C, without a penalty and a lot of legal scrabbling.
Having no idea what the realistic business value of working with C is, it's hard to give much advice.
You could lawyer up on this, and maybe make some progress. By law contracts are supposed to be mutually beneficial to both parties. There may be other laws and such in your region that prevent you from signing away certain rights, and override this contract. This is most likely an expensive pursuit, so you should be REALLY sure of the value of C.
If B is a corporation and not a person, what are the chances that they go out of business? (that would pretty much invalidate this as well).
My gut feeling is that this is probably one of those expensive lessons that you can't help but learn in real-time in the world, and you should probably do your best to push back, but not waste too much time/effort fighting it.