Could you please clarify your first sentence?
"I've found a 1903 varant that is not marked at a Springfield 1903 marked with the made in 44 on the barrel."
Is the barrel not marked, or the receiver?
If the barrel isnt marked, it may have been rebarreled, and you would need an original barrel to get it back into original condition. The barrels were 24". Many were cut to 22" when sportered. As mentioned also, if it was a standard GI 1903 or 1903-A3 drilled and tapped for a scope, then the original condition restoration isn't practical. If you just want to make it look more original, and leave it scoped, then restocking it with an original style stock isn't a bad plan. One piece scope mounts (Leupold, Buehler, or Redfield) can be had if it doesnt have one, and low power Weaver scopes like the K-3 make decent stand ins for the original scopes used on sniper type rifles. Someone was making original type mounts a few years ago, and a scope to match for the period. It would make up into a fun and interesting shooter.