It will require a rethink. 22lr has been around for 150 odd years, and ammo made today needs to OK in old rifles.
No reason why you cannot make a monolithic copper bullet that will open at subsonic velocities. Nor any reason why it shouldn’t be accurate.
Peregrine and Discreet Ballistics have developed such bullets for us .308 calibres for bigger four legged and I suspect two legged animals. And no I sm suggesting that a 150 gn subsonic .308 bullet out of 300 Blackout is needed for rabbits or fox.
Now if there was .224 or .25 diameter 40 to 50 odd grain bullet using same tech, being shot out of 1 in 10” or 1 in 8” barrel at subsonic velocities you would have a silent effective varminter. As for cartridge - well you wouldn’t need the volume of the .223 Rem but you can do subsonics with bulky low energy powder. You could of course do a 22 Long Rifle version, but that need to be a rimfire large factory type set up which would require a big player to see the market potential. And it would need a much faster twist rate.
Given all the challenges with fundamentally changing a cartridges characteristics with both Saami and CIP (basically once cartridge is listed it can’t be fundamentally changed), a new cartridge may need to be introduced.
Just like there have lots of positive developments in both rifles and shotguns no reason to think that a subsonic vermin and small cartridge solution cannot be developed.
In the meantime many controlling vermin have switched across to the 17 HMR with non toxic bullet. Yes they make a crack but its not that loud. And to be honest driving around in the dark on a quad or pickup probably disturbs more than the cracks of a moderated .17 HMR.