Short answer is "I don't think so".
Although the current SAAMI specs for the 7x57 Mauser provide for a maximum working pressure of 51,000 PSI (this was changed in the 1980's, as I recall, due to the proliferation of modern rifles in the 7x57 chambering), I'm quite sure that the original 7x57 Mauser was designed to operate at 40,000 PSI or less (the 8x57 is still set by SAAMI at 35,000 PSI).
The .308 Winchester, on the other hand, has a working pressure of 62,000 PSI. 7.62x51 NATO (what your rifle was re-chambered to) works at 50,000-52,000 PSI, IIRC. So, it would represent a pretty harsh load for such an old action (like a "blue pill" or proof load). Feed her a diet of that and lugs get set back, bad things happen.
I can't say that I'd recommend shooting .308 or 7.62x51 in the old '93 Mauser.
If you want to load light loads for it, be mindful of the pressures. Most of the manuals that provide "modern" loads for the 7x57 include a warning not to exceed starting loads in the old Mausers, for this very reason.
.308 Winchester ammo exceeds even "modern" 7x57 pressures, so be warned.