Mine is also a Sony Xperia 10 III.
Build 5.0.0.72
Device adaptation 1.0.1.48
Kernel: 4.19.248
I use the method described in the following post to give AAS control over Bluetooth: Potential Bluetooth Support in the Android App Support (AAS) - #206 by rdomschk
You also need to tab repeatedly on the Android version in the settings System → Android AppSupport to get into the android settings to pair Bluetooth devices.