From the description, it sure sounds like you already have a wireless doorbell that's ready for a battery replacement.
No battery in button. Still no unit found in house.
I suppose I have a tendency to overthink solutions to situations I have never been in before. Plugging "wireless doorbell" into eBay only gave me THIRTY-TWO THOUSAND auctions. As I am the analytical type, I must review every auction to make a decision. May take a while. In the meantime, all of you may continue having a guffaw at my expense. No, really, be my guests. It's the least I can do.