Never argue long with some dealer pinhead trying to jack you around.
Option 1 is, you go in, and tell them 'This is supposed to be covered, and I want it fixed. If you don't cover it, I will go out and buy my own battery, and it will be the largest Interstate battery I can find that fits, which will be far more expensive for you than this one, I can assure you. After that, you will receive a demand letter for the full purchase price, installation, and tax.' Make sure your letter says, "SUBJECT: DEMAND LETTER", and address it to "ATTN: AGENT FOR SERVICE OF PROCESS, (XYZ DEALERSHIP NAME)", ie their legal representative. Most likely he will realize you have some knowledge of law, and a case they can't possibly win. They will direct some minion to pay it immediately.
Option 2 is even more fun, but probably only if you are a relatively big dude. Make a big sign on foamcore posterboard that says, "MY WARRANTY SERVICE DENIED BY BUD LONG CHEVROLET" or whatever the dealer's name is. Put it on the back window of your car and roll on down there on a Friday night and then again Saturday morning. Park right out in front and sit on your trunk with the sign clearly visible. You will cut their floor traffic in half. Some people will come talk to you first, before setting foot in that dealer. You can also paint it on your car with tempera paint that will wash right off later. If they come out and want you to leave, tell them when the battery is fixed, you'll think about it. If you are small they may threaten you, so be big or at least a black belt.
Option 3 is appropriate if you and others are being truly and seriously cheated. Hang out in their service department during a busy period -- see if they are doing stuff like this to anyone else. Then find an attorney willing to file a contingency class action on behalf of individuals denied valid warranty work at all area dealers of that brand. My guess is the dealer will settle out of court for an undisclosed sum most amenable to you and your attorney just to make that sħit go away. This kind of stuff is terrible publicity.
PS: Learn how to ask for refunds - be fair but firm. I have gotten money back from multiple mechanics in the past when they knew they screwed up. One was for totally unnecessary repairs at an independent garage ...$385. Another one put a new alternator in backward on my old Plymouth and wanted another $400 more to replace my speed sensor when my battery died the next day! Instead I took it elsewhere for a second opinion, got it diagnosed and fixed, and presented the manager with the other garage's $100 diagnostic bill, which he readily paid. I also demanded and received over $500 from a major national home center chain for their incompetent actions related to my living room remodel.