Noticed the thread , thought I'd contribute.
Used to mount scopes all day long at retail level, probably mounted over 500 rigs. Not all rings and based are created equal as far as machining. Eliminate that first, try another brand. My advice, Talley, and his QR rings. This will give you easy access to your irons as well. Second, the scope type does not matter big or small, however, I would go with something smaller on a 375. Once you try some Talleys if you are having the same problem, try lapping , the first or second stroke will tell you how "off" your level you are. If its extremely far, make Winchester screw in a different barrel, been selling factory rifles for years, sometimes they just don't get those barrels in as straight as they should. A competent gunsmith could fix that as well but why not make the factory fix their mistake.
Good Luck-