Hickok45 did a video of shooting subsonic .223, surprisingly it looked like it was still punching holes in a steel drum.
Yea, rimfire is cheaper. But if you want to build an upper shooting subsonic .223 I'd say a 16" barrel with a carbine system or a 12-14" w/pistol length system would be a good start.
Gonna need a pretty good sized gas port so I'd say an adjustable gas block, a lightweight carrier and a softer buffer spring would be good too.
I've not seen anyone actually build one from the ground up so I can't say what will absolutely work.
But I can say for sure, subsonic .300bo is WAY cheaper than subsonic .223, and there's lots of inexpensive .300bo uppers.
FWIW I have a Ruger Mini 14 in .300bo, it won't cycle subsonic unsuppressed but works fine as a 'straight pull', that is I can manually cycle the action between shots without jamming the action with a spent case. So maybe a side charger would be a better option, so if there's not enough 'oomph' to cycle, you can still do it by hand.
Just out of curiosity I'd thought about getting a box of subsonic .223 and see if it would cycle in my 10" AR pistol if I dropped in the lightweight BCG & softer spring from my .300bo pistol.
But Atomic subs are a buck a round for a 50 round box and while Detroit Ammo has theirs for half that, they never have a 30 round box in stock, smallest size available is a 120 round box.
And if subsonic .223 is your absolute goal, You may find something like a bolt action .223 is an easier solution. The Ruger American Ranch even takes AR mags.