Many years ago, we, my father and I, had a 722 built into a 6mm-222 Rem.
My Pop was the inspiration because he shot Kangaroos commercially and didn't like the 223 cal for Kangaroos.
All was well with that wildcat, except it was VERY lacklustre and didn't like heavier bullets than 70g from memory, I was only 10 at that time.
Today I run a 222 Rem in a Sako L461 'Vixen'. Brass is hard to source, way pricier than it should be, but lasts many firings.
If I understand you correctly, a 6mm-222/223 will be a nice wildcat, but just be aware that brass is getting harder to source.
I also have another Sako L46 'Vixen' in 222 Rem Mag, haven't seen that brass listed for 10+ years.
Good luck with your build.
Cheers.