If you are going to modify, here's what I would do...first get original rim for rear. Replace the seat and bars with period correct Schwinn banana seat and ape hangers. Be a cool period correct custom and great to ride. Those banana seats and bars let you stretch out a bit. Keep all original parts, of course, and you could put back to stock in ten minutes if you wanted.
Good news on the rim, it is in fact stamped SCHWINN I misread it in the sun, I like the idea of the Stingray set-up, the white grips could still be used and a white seat would be cool. Will these rims fit 24X2.125 tires? I have the wider tires on my other 26" Schwinn,s and I love the cushy ride they give.