The Mosin Nagant has been altered to use the .30-06 cartridge in the past, by setting the shank back about 1/2" or so and rechambering, so the action can be made to handle rimless cartridges of that head size.
The complete alteration requires some straigtening of the sides of the magazine box to provide clearance for the longer case body.
Those rechambered MN seem safe enough, though some are leery of them because of how far the new chamber cuts away from the reinforce.
A new barrel should not cause those concerns.
The Finns prefer to rebarrel or rebore for wildcat cartridges based on the 7.62X54r case. Theres quite a line of these wildcats and some are available as factory loads, but they'd have to be imported.
You might find a replacement Mosin Nagant barrel , I've heard these are still to be found.
MN replacement barrels have been fitted to the French MAS 36 which has the same diameter and threads as the MN.
I wonder if the .348 Winchester or similar rounds would work well in this action.
Since the design is optimised for a rimmed cartridge theres quite a number of rimmed cartridges that might be better for the action than more common rimless cartridges. Belted case cartridges are another possibility, and should fit the boltface.