Except that I tried
forgetting (unpairing) - pairing again,
downgrading AVRCP to 1.3 (allegedly it helped some people, but not me),
restarting both the phone and the car,
and nothing worked.
Not only that I had an unstable Bluetooth connection in the car when using Poweramp to play music (if I used Google music everything was fine, provided I hadn't started Poweramp before it), but I also couldn't make or receive calls via Bluetooth, or update the contacts list or speed dials.
Today I decided to try the drastic step and I uninstalled Poweramp. Boom! Everything is working now - I updated my contacts list, I can make calls and I can play music (sadly on Google music app which is far inferior), I didn't even restart my phone.
So I am very much inclined to say that Poweramp is the problem. It could be that my car doesn't play well with Poweramp, but since the same kind of problem was reported for a bunch of different cars, Poweramp remains as the only common factor.
I love the app, I've been using the Pro version for years now. I hope it gets fixed soon.
Android 9 (1 May 2019 patch)
Toyota Prius V