there are many. Many many .222 bullets out there...they are not made for .223 barrels though.
OP, I’d say it’s worth a shot, but I wouldn’t push them very hard. Lots of guys shoot .30 caliber bullets out of Mosin’s typically have bores closer to .311/.312. I can’t say I’ve ever tried it, but it’s relatively common. THAT being said, the difference PERCENTAGE wise from a .224 to .222 is steeper than a .311 to a .308, so maybe do some Google fu on it and see what you come up with first?
One ‘hang up’ I can see is in bullet seating though...a .222 bullet might slide right into a case sized for .224.