Cheap device that can sniff BLE and send it to Wireshark:

https://www.adafruit.com/product/2267 ($20)

Overview on using it as a sniffer: https://learn.adafruit.com/introducing-adafruit-ble-bluetoot...

Still a lot of work though, if it's anything like LED signs I've reverse engineered. Trying to figure out if it's length encoded payloads or delimited payloads, what obscure checksums it might use, weird encoding of images, etc. And experimenting often hangs the device, or changes it's behavior significantly.

Not reverse engineering, but I'd used free iOS/macOS apps that could inspect the traffic content. I also had access to the spec sheet from the manufacturer ("smart" BLE scale) that had where in the long payload digits were the weight, body fat, and water content, etc. Kinda fun and frustrating at the same time. If you truly wanted to punish yourself, you'd be doing this while trying to write a React Native app that communicated with the scale over BLE. :)

Using something like this[0].

[0]: https://greatscottgadgets.com/ubertoothone/

If there is an android app for the device, you can easily log the BLE data and load into wireshark

https://www.mybluetoothreviews.com/what-is-bluetooth-hci-sno... shows how to capture the packets (you just go to the developer settings to turn it on)