My parents bought a 2006 Cardinal 34TS, which is 37' long with a GVW of 14,410 lbs., and it has 6k lbs. axles. It came from the factory with aluminum wheels and Kenda Karrier ST235/80-16 tires rated to 3,500 lbs.@90 PSI. (Also, the aluminum wheels are rated to 3,100 lbs. @ 80 PSI.) I replaced those tires in 2011 with Bridgestone Duravis R250's in the LT245/75R16 size. After some local miles and an uneventful (at least tire-wise), ~3,000 mile trip last summer, we now have about 3,500 miles on the tires, and we have had no problems so far. I'm not promoting anything, just conveying my experience.
Maybe I misread the long tire thread started by Capriracer, but I recall it being suggested that ST tires be loaded to only 85%-88% of rated capacity to maintain a safety margin, or "reserve". If that's the case, then a ST235/80-16 tire, with a rated capacity of 3,420 lbs. @80 PSI, should be loaded to only 2,907 lbs.-3,009 lbs. to maintain a margin of safety. The LT tires I used are rated to 3,042 lbs. @ 80 PSI (at 99 MPH, also).
Your camping options will be determined by the weather at that time. As far as camping in the parks is concerned, you'll probably be limited to Azalea at Grant Grove in Kings Canyon (snow on ground will be determinant) and Lodgepole in Sequoia. There will be no dump stations available up high during the winter and early spring; you'll have to drive down to the mobile home park across from the Hume Lake Ranger Station to dump your tanks.
National Forest camping opens around May 31, depending on snow levels. Big Meadows is our preference, and it will be the last to open as its elevation is ~7700ft. There are five camping units in Big Meadows. There are no designated individual campsites and no piped water. We disperse camp outside the units with our 37ft 5er.
The campground in Three Rivers mentioned by some is a good choice if you mostly prefer to visit Sequoia NP. I believe it's right around 1,000ft in elevation, so it'll be comfortable.
Last year, I installed LT245/75R16 LRE tires on our Cardinal 34TS 5er. It has a GVW of 14,410 lbs. on a pair of 6k axles. I went with the 245's over the 235's because I was more comfortable with the distance between the front and rear tires the 245's would yield. Before I made the decision, I checked wheel well clearances inboard and outboard of the tires, as well as in front of and behind them. I checked the distance between the wheels and the suspension components, also.
This past summer, we took a 3,000+ mile trip around the northwestern states. I bought a tpms before the trip to help monitor the pressures and temperatures as we traveled. We didn't have a single problem related to the tires. They even performed well during a heckuva downpour in eastern Idaho. The tpms showed that the PSI and temp were very even across all four tires. (Also, the tire pressure gauge I carry in the pickup was within ~2PSI of the tpms sensors.)
For the trip, my GCW was ~21,500 lbs. with ~10,700 lbs. on the axles. I don't know the individual weights of the TV and 5er as I didn't have a chance to separate the combo.