I am assuming you're using Gnome as your desktop environment. Similar options might exist or not exist in other environments.
I solved the problem described by you on my Ubuntu 20.04.5 LTS several weeks ago and did not yet upgrade to 22.04.1 LTs - but checked the changelogs.
It is more than probable that your HiFi Audio Device supports more than one Bluetooth profile - so do most of the recent BT headsets.
A general word about two-way Bluetooth-Connections: There is no "non-proprietary generig" way (or at least I don't know any), to have a two-way HighQuality audio connection via Bluetooth. Device Manufacturers might supply proprietary, device-specific software to accomplish this. In most cases this software would not be compiled for Linux.
One possible workaround could be a Bluetooth USB dongle being recognized as a (two-way) sound card by your Computer. But this is beyound your original question.
using your HiFi Audio device to play sounds coming from your computer
Once you have connected your HiFi Audio device via Bluetooth, go to
gnome-control-center sound (either by klicking through "settings" or by running that command after pressing F2 or Ctrl+Alt+T (3 at the same time)).
First scroll down to "Input" and set it to something else but not your HiFi Audio device connected via Bluetooth - or mute it. Either will work.
Second go up to "Output" and choose your OutputDevice (obvious) and the Bluetooth profile of your choice beside Configuration (A2DP Sink with one of the supplied Codecs)
On Ubuntu 20.04.5 LTS I had to do an additional step before this became possible:
Unfortunately the above-mentioned package's "activity" looks like a one-time shot packaging https://github.com/EHfive/pulseaudio-modules-bt for Ubuntu 20.04 LTS.
The version of
pulseaudio found on Ubuntu 22.04 should be able to support this "out of the box" - according to the changelogs I read (did not test with a live ubuntu 22.04.1). If not - find a suitable ppa for 22.04 of
pulseaudio-modules-bt - chances are high.