In future discussions about this situation don’t say “they put my battery in backwards” just say they "hooked my battery up backwards’ I say this because I really had to sift through a few sentances in your post that really did not have to be there.
Get your story down real clear and don’t “gusssie it up” with things like “an experienced mechanic came over” just say they hooked my battery up backwards and they admit it. If I was in a position to make a decision about your claim I would want to just hear the cold facts.
I jumped a battery backwards once(I was real young) it sounded like a .45 went off, blew the tops off the battery and split the case, very dangerous.