Several things come to mind, of which a couple have been mentioned. The 7 RUM is a fire breathing dragon. I know that with my 7RM the second shot gets the barrel warm and the third shot will get it hot if I dont let it cool long enough. I would think that with the RUM, after the second shot, that barrel is very warm unless you have let it cool between each shot. Combine this with the fact that you are shooting a factory barrel and sporter contour ( I assume it's not a varmit or Sendero contour) that probably has stresses in it and a little change in temp may affect it's dynamics quite a bit. The HS stock probably doesn't *need* bedding, but in this case, if the stock wasn't custom fitted to the action a little skim bedding might help and probably wouldn't hurt if done right.
Also, cleaning after each 3 shot group might not be a good idea if you are shooting for groups. Your factory barrel might need to be fouled with one or two shots after cleaning. You might try cleaning the bore and then shoot a 10 shot string, letting the barrel cool between each shot, and see how the first couple or three shots group compare to the rest.
Being a factory barrel, I would especially give the barrel a good break-in, cleaning after each shot until it stops or almost stops copper fouling.
On ammo selection... if you are looking for a high degree of accuracy, you should be hand loading with quality dies, bullets, brass, and powder. Use the core lokts for break-in.
Berger VLDs coming out of a 7 RUM are going to be highly explosive at short to mid ranges. My first choices for bullets would be 160 AB's and 150 E-Tips, in that order, which ever shoots best.
Just my thoughts, hope it helps...
-MR