Sorry ahead of time but I can't help myself ... Why not get a Tikka Superlight in 30.06, send it to J.E.S. to have him make a 338-06 or 35 Whelen, cut the barrel to whatever length you fancy and "have your cake and eat it too" - inside of 300 yards you wouldn't be giving up much delivered energy ...... ( I'd choose the AI in either, not because of performance but to make brass a bit easier to work with)
As good as the 338-06 and 35 Whelen are....they are certainly not in the same class as the .375. In most loads the .375 is going to carry 25% more energy and has more frontal area.
In N. America you're unlikely to need such a cannon- but after shooting several critters with both .338WM and the .375...the .375 is certainly a more dramatic killer.