Just ask them one simple question:
If I had brought my car in for warranty repair, would you have fixed it?
If they say no, the warranty paperwork wasnt sent in so you didnt have a warranty, then remind them that the check you gave them did in fact clear, so the money was real.
If they say yes, we would have honored the warranty, it was our oversight, then off the top of my head I dont know how to proceed from there.
What you are buying with the warranty is peace of mind, since nothing broke, and you were under the impression that you were under warranty for the last two years, one could say that there was no difference as far as your experience (aside from the aggrevation of getting it straightened out now).
I'm no lawyer, but I think to win (like in court) you have to have a few things, duty to act, breach of duty, damage and proximate cause. The warranty was their contract, it said what they had to do (duty), the car didnt break (so they did not in fact fail to fix it, that would be the breach of duty), you didnt have to pay for any warranty covered repairs in that time (which would have been damage to you) and since we have no damage you cant proove that their breach of duty was the proximate cause of the damage you didnt suffer
In all likelyhood, if something did break in the last two years, you would have gone in and they would have come to the same conclusion (hey ...this guy did pay us for his warranty) and they would have fixed it (both the car and the warranty problem back then).
I hate to rain on your parade, but if I call it as I see it, you are probably entitled to the rest of the warranty for the period you already paid for.
GM is a very big company, I'm pretty sure it wont break them either way. If you wind up with another three years,.....good for you
Good luck.