What year Mustang? Why is so tight? Are you also running a engine-driven fan?
You could always get an electric fan that acts as a pusher for the front of the radiator. They aren't as efficient, but the right one can be enough.
I've used those straps on a copper radiator before, the trick is to make sure they stay tight. The same would apply for the aluminum one. If the fan loosens up and starts moving a bit, that's when it'll get ugly, but shouldn't fundamentally hurt anything but the look of the cooling fins around the straps.
1969 Mach-1 Mustang
460 / C6 / 9"
Back in Dixie DSO 22 (Charlotte)