Les, I have one of these rifles in .458 Win. that I have been playing with since last summer. I intended to put two crossbolts into mine, but thought I would try a thorough glass bedding job first. After about 150 rounds of fairly stiff loads, it seems to be holding fine without the bolts.
I would recommend that you carve out some space under/behind the rear of the tang to build a good thickness of glass, and either use tape on the back of the tang, or carve out some relief in the glass at that point so that the metal does not contact wood at that point. The recoil level with this particular rifle in .375 is not bad at all. I believe that glassing all the stress areas will take care of any stock splitting issues.
Oh, one other thing that experience has taught me with these rifles is to get that front sling stud off of the bottom of the fore end. It can and will bite you. There again, might not be much of a problem with the .375, but it made for a painful afternoon with my .458. Relocating the stud to the front end of the fore end works just fine.
Best of luck with your new rifle!