How quickly does the battery overflow when you charge it? You do not want to let it push out any fluid. Set the charger on the lowest setting, and charge it for the short time that it will take it before overflowing.
Can you get the charger powered at the truck, to help start it? If so, when you have the alternator and cable finished, use the charger to help start the engine.
I agree about the battery warantee, if you can do it have it exchanged. Tell them simply that it doesn't have enough power to start the truck, and/or won't hold a charge. Skip any mention of freezing, the warmer malfunction, or the alternator. They will have the tools to check it, and to charge it properly. Let them fix(charge) it or replace it. Good luck,