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Your tire store is right. My period Tire and Rim Association manual shows a 235/75R15" tire is designed for a 6.5" wide rim. The narrowest approved is 6" and widest approved is 8.5".
There is nothing wrong with riveted rims as long as they are sound. One of my tire professional friends advocates against mounting radials on riveted rims, and I know the wheel manufactures had trouble with big truck riveted wheel failures back in the 60s/70s when radials were first introduced for big trucks. During the 1960s industry adopted welded centers perhaps because of this reason. You'll get a variety of opinion on whether radials are safe to use on car and light truck old riveted rims. I've not found a scientific study to show one way or the other. I know there's zillions of guys running radials on old rims without concerns. Stu
Or make a cheap caliper out of scrap to either measure around the tire or through an air vent if there are some. Measuring outside of the lips means you'll need to subtract for rim lip thickness on each side. Stu