15 x 10 tire suggestions

Well, the tape measure says he has two inches until he hits the spring. That’s a bit of tire raised off the wheel.

The section width is your problem then (Assuming you don't have a full open wheelwell). The deal is: the wider the rim, the wider the tire needed.
You might have to go to a 255, but that is quite narrow for a ten inch rim.
The 275 (accord8ng to Tiresize.com) is 10.8", but that is probably measured on an 8/8 1/2 rim, section will be wider on your 10". Call it 11.1or 11.2, you will have a sidewall bulge toward your fender of about .6" .Very tight, might work.
What body style, and how high is the rear?

72 Dart Swinger, stock openings. I recall Chip Foose had a clearance problem on one his cars he did, he used a piece of wood between the tire and wheel opening to pry a little relief