In addition to my K1200GT, I've got a Goldwing 1800. I put a Michelin Alpin run-flat tire (which is actually a snow tire) on the rear of the Goldwing and have been happy with it. However, I don't know that the K1300GT has the fender clearance required. You really need to measure that before making the plunge. My Michelin barely fits (deflated) and mounting it was a challenge at the motorcycle shop (who was promised anonymity and was paid in cash, no receipt). You'd also need to know which tire mates with the rim of the K1300GT. There are enough Goldwing users that I had plenty of advance advice.
It also depends how you ride the bike, too. If you're using the K1300GT as a canyon carver, then you might be happier sticking with a conventional sticky motorcycle tire. I use my Goldwing as a long-distance cruiser, mostly on the super-slab, where the car tire is at home. When I replaced tires on the K1200GT, I went with a motorcycle tire. The car tire requires a little more tip-in pressure on a turn but I've never felt it was a problem. You've probably seen the Goldwing rider with a car tire doing the Tail of the Dragon in Deal's Gap way faster than I would dare.