Instead of contacting your lawyer, call the Better Business Bureau. Tell them what happenned, accurately, and they will tell you what to do. I think you would be putting SO much energy into this and time, just to get your 360 repaired. But, in all honesty, were you doing the "right" thing trying to get a warranty on your 360 AFTER it had broken? Isn't that like trying to get Flood Insurance AFTER your house has flooded?
Either way, I'd say you're in the wrong. They made a mistake. Documentation says that you have to apply within 90 days. You did not, therefore the transaction you made should be null. It shouldn't matter that they were going to give you the warranty since you yourself did not comply by the rules set forth in the warranty. It was over 90 days.
After you are done arguing this (feel free to take it as far as you want, I just don't think you'll get far for a LOT of energy), maybe try replacing the Thermal Interface Material with some Arctic Silver 5, it might fix up your problem and save you a lot of hassles.