You'd have to do some guesswork though to figure out what's going on but it would be better than nothing, and you might be able to get the size down below what CR2032 allows.
If you want it to be compatible with most smartphones, you'll need Bluetooth LE anyway, and that already uses at least 10 bytes on header fields for broadcast packets.
I wouldn't be surprised if there was some trivial way for the manufacturers to add ultra low speed non-bluetooth data to the Bluetooth chips though, so you could have 2400 baud IDs with microwatts of power.
But it's probably not worth it since I suspect the real power drain on airtags comes from the high update rates and maybe the 2 way communication.
Why? If you're using your smart tag to track your car, you can hide it somewhere deep inside and connect it to the vehicle's battery so it always has power. Then you never have to worry about changing batteries.
Well, then you could only ever track 255 different tags :)
AirTags are already pretty minimal. Their battery lasts about a year, and I imagine an open source equivalent should be possible to build for under $5.
This is purely guesstimated based on the fact that I have a couple of Bluetooth LE thermometer/hygrometers that cost me about $3 a piece that last about a year on the same battery (CR2032) – and these run custom firmware and need to run a display and data logger (to on-chip flash) as well!