The new shoes will likely be Cooper STT Pro’s in 35x12.5r17 or 315/75/17. I’ll also look at Baja Boss’s, but if I find a 36” I’ll look at that. Right now my mind is occupied with the auto swap, but if I see a sale I’ll be selling my 35’s on 15’s and moving on to 17’s. I believe @kiwi was interested in them. I think you’ve given him some parts IIRC!
Since you drive yours on the road more I won't recommend more aggressive or quick wearing tires. Ryan seems to be very happy with his Baja Boss tires.
Yes that auto swap is going to be a game changer for you. You might need to be looking at new gearing once you do the auto swap... Take a step back and breathe, that auto swap isn't any harder than anything else you've undertaken. Your best thing you've got going is that you have an example sitting in the driveway. Also take your time.
My only recommendation would be to possibly look at replacing all the plastic looking with the cloth loom from Painless Wiring. I really like working with it a lot more than the plastic looming.
Only 36" tires I know of are Interco/TSL which I thought were a no no for you? I've had good luck with my TSL tires over the years but I know you had balancing issues.