Got the tabs out, but now the bottom is so clamped tight that I can't turn it. I'm afraid to try and snap the tabs... don't have a tool to do it with precision, afraid I will scratch or dent the front of the cart. Any ideas on how to snap them? Or do I just need more brute strength to start turning this thing?

The hardest part is done.
Hold the game the right way up, look at the back then try twist the back piece to the left, in an unscrew motion. If the back hasn't been melted onto the front, you might be able to just unscrew the entire back from the front. Once you have enough space to access the tabs properly, use some pliers to break them off, it would likely make the job a hell of a lot easier.
If you can't do that then I can't see any other way than putting a dremel to it, then using pliers to break the melted bit front the front, just enough to try and get the screw out. Off the top of my head, I can't remember which way the chips face. Do they face to the front or back of the cart? If they face the back of the cart, be really fucking careful with the dremel.
Once that screw is out your good. You could then go tidy any melted bits left over with a file, if it really bothers you, just be careful not to damage the thread or the screw won't go back in properly. Just literally a few runs over to make it look alright.
SIDE NOTE: Is it strange that I want a game with the same problem as your, so I can fix it?