I would start with a 223. Doesn't matter how metel they have to bore out, and it already has the rimless extractor. And you might find that the 8x57 reamer would fail to clean out the 308 chamber (at the shoulder), just like the 7x57 reamer fails to clean out the 7mm-08 chamber (at the shoulder). If in doubt, drop a dime to a gunsmith. They can give you the right answer for sure.
Wally