I don't know of any indoor ranges that can handle .375 calibers as most only go up to 300 win mag. I have never seen any non-mercuric ammo (required indoors) available for indoor shooting either. I have several .375's you could shoot but I am currently living in LA.
If you shoot 12 ga. shotgun turkey loads you are already handling more than the 46 lbs recoil the .375 generates. The 30-06 is closer to 35lbs. (someone correct me if I am wrong) with the .308 being less. Regardless of what it is you must simply get used to it by shooting yourself. If you are trying to get used to how a particular rifle feels in you hands that can be accomplished by going into a gunshop and hefting a few.
Too bad the shop on Howard and 2nd is gone, they always had lots of rifles to choose from.
Frank