If you'll accept that I'm being serious and not sarcastic, I'd suggest looking into a good .22LR conversion for it. Reason I say that is that unless you find a load with an extremely heavy bullet, that's what you're really ending up with anyway - a low-velocity, .22-caliber projectile that's weaker even than a normal .22LR round.
I've used one (a ciener) on a suppressed carbine for quite a while, and after the break-in period, it actually became my squirrel gun for a few years. It was surprisingly accurate and phenomenally reliable with remington subsonic HP's, and stunningly accurate but not particularly reliable with Aguila's heavy 60-grain SSS load. I was worried about lead buildup in my suppressor from using non-jacketed LR bullets, but AAC assured me that as long as I didn't run more than a couple hundred or so of them between sessions with full-power ammunition, that the pressure and heat from the full-power stuff would keep any buildup blown out.
If you must stick with subsonics in a full-size .223/5.56 cartridge case for whatever reason, I have no idea what to suggest; sorry.