I can't help you with tires, as I simply went darkside (car tire rear).
Many riders will say it's never worth the small cost savings to patch a motorcycle tire, and to always replace with new for the safety and peace of mind. And they'd be correct.
I have had good luck with removing the rear wheel from the bike and taking the wheel alone to an independent car tire shop (you know the ones with mountains of used tires piled out front) and having them install a T-patch from the inside. Did that with a new tire that picked up a nail with less than 100 miles on it, then ran that tire down to the wear bars without issue.
It's your bike, and your choice.
The BMW Navigator IV, V, and VI GPS units will fit in the GT's factory GPS cradle position and allow use of the handlebar Scroll Wheel. These will also fit the factory GPS cradles on the GS bikes.
The Garmin zūmo 660 or 665 will fit the factory cradles, but will not respond to the Scroll Wheel.
Otherwise, you can use any GPS by coming up with an external mounting location.
Sometimes owners over at K1600forum.com
have used handlebars, GPS units, and other useful parts available in the Classifieds section
You may also get some good info in the BMW K1600 Suspension Setup, Tires and Wheels