This bug ticket suggests you could try to disable Bluetooth autosuspend by adding kernel option:
rpm-ostree kargs --append="btusb.enable_autosuspend=0"
(Do not forget to reboot after this modification.)
You can read current value like this:
cat /sys/module/btusb/parameters/enable_autosuspend
On my Aurora install it seems that autosuspend is by default enabled.
If adding kernel option does not help, you can remove it by running
rpm-ostree kargs --delete="btusb.enable_autosuspend=0"