I'd be concerned that with a FMJ bullet, after swagging the jacket would spring back more than the lead core, leaving the core loose. It also seems it would take quite a bit of force to do this, requiring a substantial press.
Even with lead bullets, sizing more than about .005" starts getting difficult and you start having to do it in steps with multiple dies.
Sounds like a lot of effort if it's not absolutely necessary.
Unfortunately, Lee doesn't make a push through die for .222". You'll have to find someone who makes custom diameter bullet sizing dies. I found one via google "Buckshot's" Custom Sizing Dies - Mountain Molds
....
To answer the original question, yep you can swage .223" bullets to .222". I have sized down some .323" bullets to .318" to fit an Enfield I have. I used a "custom" Lee push through die and lubed the bullets with STP. It was a job, but I wanted to handload some jacketed bullets for my rifle. with a .318" groove diameter and could not find any commercially.