Here, in Canada, there are re-boring services, but re-barrelling is usually much less for the contoured barrel blank. Chambering and threading will bring the costs to about the same level.
: I'd sell the .375 and buy a new one in .400cal. Green Mountain sells .400 bore/.408 groove dia barrels through TOTW for les than $200.00. These are ready for chambering, threading and finishing.
: I am presently awaiting word from Track on the availability of a new Winchester taper Octagonal .50 cal with 1 in 24" twist for my Sharps action. They make them in most bore sizes from .375 to .50. Since theya re 4140 gun barrel quality steel, they should be just fine for any reasonable smokeless rounds, but check with GM.