Thanks for letting me know. I am seeing these issues, too, and have reported them to Apple (FB12550887 for connection and FB12550939 for the icon). If you would like to record a Bluetooth log right after the problem occurs and send it to me (via e-mail) I can attach it to the bug report.
If it helps I don’t think this relates to the Sonoma beta. I’m on 13.x and TF stopped working with my AirPod Pro’s about 5-6 weeks ago I think? Let me know if you’d like a bluetooth log.
I’ve not heard of any problems with macOS 13.x, and I use it there regularly myself. Is it reporting an error for you? Please send in a diagnostic report and a Bluetooth log.