Once converted, the rifle is no longer an M1903 & FWIW I concur, the magazines are cartridge specific.

As far as I'm aware, Steyr, Daimmler introduced a .308 cal. in 1952, into what effectively was a similar designed rifle, but not quite the same as original MSch rifles.

These older rifles with split bridges can be scoped and there are a number of different ways to fit whatever mounts are appropriate to you. I've just done that very thing and the detailes are under a recent different thread on this website.

My advice is find an un-altered MSch stutzen/rifle.
Cheers, d