If you're starting with a 03-A3 action, life will be a lot simpler if you stick with the 30-06 or 35 Whelen . Feeding gets tricky with the shorter rounds and headspace problems might crop up with 40 Whelen. I don't know about 375-06 headspace , but this is a Wildcat for sure.
My first sporter was made from a 1903-A3 surplus rifle , it came with a 30-06 Remington two groove barrel , it shoots cast boolits very well and I've never regretted sticking with the 30-06 chamber.
Parts, triggers, floor plates, safeties, were always made for this action and chambering. Feeding has never been a problem. And if I must buy factory ammo usually 30-06 is available.
Glad to see sporters still hold interest , did mine in late 60's....bought the military surplus rifle for the outrageous price of $59.95....that was a while ago!
Gary