I've played around with subsonic 223 just for fun back when TrailBoss wasn't impossible to find.
Is there a specific reason you want to go with 223 vs. 22lr? I assume it's the 77gr projectile, but in my experience a 45-55gr projectile is more than enough for rodents/squirrels etc and most standard velocity 22lr will be subsonic out of a SBR.
I found it was extremely finicky trying to get heavier projectiles subsonic with 223 due to the low case-fill and the line between supersonic and squib became extremely thin. 55gr projectiles on the other hand were much easier to consistently load in the ~950-1000 fps range but now you are back at 22lr performance just more expensive.
edit: If you aren't totally set on 223, I had a lot of luck loading subsonic 300 blackout with 110gr vmax projectiles and TrailBoss. They are also comically quiet and pack a decent punch at 1000fps if you are set on something heavier than standard 22 loads