As a former fulltimer (nearly 15 yrs/400,000 towing miles) pulling a 40' 22,000 lb fiver, I've purchased many new tires for my rig and have always used a tire store local to wherever I happened to be at the time. Can't recall ever having a shop refuse a sale/installation though I will confess that I always checked out the shop as far as ample parking/service area before I got a quote. I won't bother you with the tire brand etc since it was a fiver but post this part only because my fiver was larger and heavier than most class C's and it was never a problem to get service.
I now have a 30' Class C and last month replaced 4 yr old tires with just under 20,000 miles with a new set of Firestone Transforce. Easy tires to do a price comparison on and, as with the fiver, getting service was no issue. I trust the major chains if for no other reason than they probably wouldn't become a major chain if they weren't trustworthy as to consistency of price and service. I do watch as they dismount and mount the tires on my aluminum wheels just to make sure they don't scratch them.