Both .30 carbine and 30-06 are the same bore size. .22 LR and .22 centerfires are different bore sizes.

Will it work - Yes. Is it the best - No. Apart from accuracy issues, you'll also have a leaded barrel to ultra clean before putting .223 through it again.

Why not do the job properly and simply buy a .22 LR, or .22 Mag, barrel for it?