If you were a cop, what would you do?

They're not going to do much, at all. I think, worst case scenario, they'll just have you give her the money. They won't arrest you or anything, because you didn't steal the battery. They might even not pay her mind. But assuming they do, they can't do much. In fact, if you really feel stalked, she'll get in more trouble than you. Either way, just try to pay for the battery; if you refuse, it'll just be troublesome. She's obviously completely blowing things out of proportions though.