I bought a Winchester .375H&H some years about (back in '02 or '03). It was a stainless synthetic, not a safari model. Anyhow, I also bought a McMillan stock and had a gunsmith put the barreled action in the McMillan and do some other work. The first thing that the gunsmith said was "that's a pretty light stock, are you sure you want it in there?" and I said "yup." Well, that has turned into my dream gun. Why? Well, you probably won't believe me when I tell you that with plain-jane Federal 300 grain soft-points I can essentially put 3 bullets into the same hole! (BTW, I did have the action glass bedded, trigger job with pull at 3 lbs - you don't want it too light on a .375H&H, polished bolt/receiver, and whatever other work that I forget about.)
Anyhow, that stock has help up very well, and I have shot quite a few hogs with it. (I got tired of seeing hogs run off even with well placed shots from a .30-06, so I got the .375H&H. Now they just drop in their tracks.)
I really can't comment specifically with regards to a Remington BDL or other wood stock. I can only generally say, or at least think, that it might depend on the actual blank of wood being used, as sometimes you may get a kinda crappy part of wood and not know it, whereas with synthetic stocks, there is more uniformity with production. However, that been said, I would think that your standard Remington wood stock (with proper channeling and other minor fitting adjustments) should be able to withstand the recoil from a .375H&H. I'm not making any promises, just my somewhat educated guess.
Zachary