@kristatos
maybe doing some more tests.
I tried it again with my phone (FP5) setup with 24.04-1.3 stable and succeeded with receiving and sending test files.
However I have reproduced some of your problems.
Not the obex service not known. This was not happening for me.
I have seen the disconnection of Bluetooth happening though. Strangely the Bluetooth page for the device (an Android phone) was displaying the device as connected BUT with no signal, then disconnected very soon after (may seem logical given there was no signal).
I removed each phone from the Bluetooth configuration of both phones, tried again and this time the signal was rated 'excellent'. It then stayed connected. Needless to say, for all these tests the phones were sitting on a table very near of each other. It looks as if the state of connection is rated with 'no signal', it's staying so even if the real connection is good. In this case the pairing had been done much before and the phones had both been off or way to distant of each other to talk with Bluetooth.
After solving this connection problem, I think another trouble may be that if a transfer is not acknowledged by the receiving end, all following transfers fail. Not sure but it looks like that. This problem was happening with a transfer UT -> Android.
If my impression is correct, it may be a problem with the app (the connection problem mentioned before is probably more caused by the system bluetooth stack).