I would guess the costs involved in any conversion of the 1903 to another cartridge, even if you keep it in 6.5 caliber would far out pace just reloading the 6.5M/S.
Re-machining each separate case form on the rotor is a challange and there is some action/rail work for a larger diameter case to feed properly.
Case /cartridge length is critical and cannot be extended past what the rotory mag will allow.
That may limit proper bullet sellection should a longer case than the original be chosen in a conversion.

Though the 6.5M/S die sets can be pricey,,they can be found for less than retail on the auction sites.
100 new cases will last a long time if you neck size after the first loading and keep loads reasonable.

Plus an unaltered 1903M/S is an easy re-sell if it ever comes to that. Any conversion is a tire kicker no matter how well done and in what caliber. The buyer will want it for less than the original.

I shoot the 6.5M/S (1903 M/S carbine) and the 6.5x53R in a Steyr sporter. I load both with the same set of dies and make the latter out of 303 brass & 30-40Krag.

6.5M/S can actually be made from 220 Swift quite easily if you get stuck for branded stuff, but there are quite a few sources around now unlike some years back.

6.5 Italian brass is the right size (base dia) but short over all,,51mm vs 54mm.
If you can fireform it w/o haveing a headspace problem while doing so, you'll have a usable but short necked M/S case.

..and you can always make it from 303Brit & 30-40 Krag with alot of work including turning off the rim and cutting an extractor groove besides the forming & trim operation.
303B have needed inside neck reaming in my experience but 30-40Krag has not. But I've only ever used Remington 30-40 brass and HXP (Greek Milsurp) 303 for the operation so others may give different results.
The HXP needed a bit of the base OD trimmed also where the 30-40 didn't. Again, different brass mfg may vary the results.

Most 1903 M/S and Steyr military rifles in that caliber (rimless or rimmed) seem to be of .265/.268 groove dia. from what I've seen. My 1903M/S is .266groove dia,,my Steyr sporter built on a Romanian 1893 barreled action is .268 dia.
I use standard Hornady 160gr rnsp most of the time. They are .264dia and give excellent results.
I have a large supply of pulled milsurp 6.5 Italian 160gr bullets also. They measure at .268 but give excellent results also.
The deep rifling helps with the slightly undesized bullets I think.

Many commercial M/S rifles and carbines have excessive headspace also,,but reloading practices can negate the issue.

6.5M/S paper ballistics aren't all that impressive, but field results especially with the 160gr bullet it was designed for are.
The M/S feeds the round nose bullet flawlessly.