Well, they're not rechargeable - you swap them out. Secondly, when you DO get the low battery warning from FindMy, it enables the battery indicator for that item, showing it has a low battery. (at least, that's how it worked on the two items I've experienced it with).
The low battery indicator disappears a few seconds after you install a fresh battery and the airtag gives its status to FindMy.
Edit: Hmm. It would be interesting to disable your cellular and wifi radios (before replacing the battery) - see if the battery status updates when you're isolated from the FindMy network.
Also, if the CR2032's weren't so cheap, I'd say get it replaced under warranty - but for less than $1, it's not worth even wasting your time talking to Apple about it.
If that airtag chews through another battery in 3 months, you may have a dud - and then I'd contact Apple to get a replacement.