The Kodiak's usually have an Accu-weight hand written on the sticker by a factory technician. I believe that weight is with everything except the propane, and full water tanks and perhaps batteries. That might give you better number to work with rather than the published dry weight.
If I recall (it was written with a marker and now warn off) my Accu-weight was 5,280 lbs, while the factory specs was 5,214 lbs. Not much of a difference. I can't see that you need more than 1600 lbs of extra gear etc...
I can't help you on the tire question, I'd be guessing. I do have the Husky Centerline WDH hitch and love it. It is easy to hook up and stays level and straight as an arrow behind my truck, even on the interstate with passing 18 wheeler's.
Good luck!