Tundras have a lower payload rating than most other 1/2 tons, I believe, due to the way they're built - heavier framing under the front, to support the engine, somewhat lighter framing under the passenger compartment, and c-channel framing under the bed. That of course is where the weight of any 5th wheel goes - on the lightest-built part of the truck.
I don't think I'd recommend towing any 5th wheel with a Tundra. JMHO. Otherwise, it's a great tow vehicle.
2nd best way, very effective: a tarp that covers the camper just below where the top meets the body (when the top is latched down). This allows some "breathing" but protects the roof from leaks. Bungie the tarp down to the frame, so it doesn't blow around. This worked well for us for about 12 years.
We travel with our fridge on propane. Back in the day, when they made 3-way fridges, we used the 12v option on the road, but that's no longer available. Frozen food does fine, as does everything else. I make a list of potential meals, from that make a list of groceries needed, and pack what will fit. With no pantry, it takes some creativity, but we can get about 5 days' meals in the fridge, drawer (which serves as the pantry) and a few sections of cupboard. You'll figure out what will fit over time. There always seem to be grocery stores not far from where we camp, or the ubiquitous Walmarts along the way.
We used to tow a 7500-lb (GVWR) 34' trailer with our 1/2 ton, and after 6 years of many long trips with that combo, I was tired of towing such a large one. (Weight was not as much as issue with us as was size.) We traded it for a 6000-lb (GVWR) 27' trailer and I (as the driver) am MUCH happier. It handles better, and we can fit into more national forest/state park/national park campsites with the shorter trailer than the longer one. I'd advise the smaller trailer you're looking at - even if you're considering upgrading the tow vehicle.
Previous comments on payload capacity of the vehicle and LT tires instead of P tires I also agree with. We did upgrade our tires to LTs a few years ago as well.
Before the current truck and the big trailer, we towed a 24' hybrid trailer with an SUV (Ford Explorer, before it was redesigned into a car) and it was not a pleasant experience. We did feel the "tail wagging the dog" even with sway control, due at least in part to the short wheelbase of the SUV. Another reason to choose the smaller trailer in your case. :)
Ditto GeoBoy's advice. IMO, Colter Bay RV Park is by far the best non-NFS/NPS CG in the Tetons. Grizzly is a lovely park, quite convenient to Yellowstone's west gate. Both have FHUs and can accommodate a larger rig.
Grizzly will be fine for a base camp for Yellowstone. There isn't much on the east side, nothing except Fishing Bridge has hookups anyway. If you don't need hookups, then Bridge Bay might work for you; some of the sites can fit your rig (though many cannot). Yes, there's a lot of driving, but there always is from any location in the park, and better (IMO) than constantly changing campsites.
5 days for YNP and 2 days for GTNP is good. A 4-and-3 mix is good as well, but will give you a noticeably shorter time in YNP to see things.
Is it actual sway (as in, you can see the trailer's back end going back & forth) or is it "wind wag", where the trailer itself seems to be slightly rocking back & forth? Wind wag is pretty common and something you usually can get used to. Could be your hitch needs some adjusting. If it's sway, then yes, the sway bar should be doing a better job.
We tow with an Equal-i-zer, which has two rigid sway bars which use friction against brackets on the tongue, rather than being attached by chains. Have not had a problem with sway, even when towing our previous 34' TT (and in pretty windy conditions; if sustained winds are over 35 mph we usually try not to travel.
What comes to mind: any site along the bluff edge in Wisconsin Ridge CG, Wyalusing SP in sw Wisconsin (still our favorite view); sites along Lake Superior in McLain SP, Michigan; and the lakeside sites in Salmon Lake SP, Montana. Some great views from almost any CG on the east side of Glacier NP, too. But Wyalusing tops them all, hands down. (The bluffside sites there have no hookups, but they're worth it. Not for big rigs, though, sadly.)
So sorry to hear that! It's such an awful feeling. :( Hope you're able to get at least some of the heirloom stuff back.
Not saying the OP did this or has done anything to promote the burglary but as much as I would like to share my adventures with others this is why I never post on social media "hey look at me, I'm on vacation somewhere". Maybe it's being paranoid but I just don't trust that information won't get into the wrong hands. Ditto. (Not implying the OP did this, just a general warning. I see it too often on social media.)
Years ago we had a small MH that had the awning on the opposite (i.e. street) side of the camper. We would have used it more often had it been on the curbside. (Entrance to the MH was on the rear of the rig.) So I would also vote for putting entrance and awning on curbside.
We spent a weeklong vacation across 3 or 4 states with such a combination one time; our tow vehicle was involved in an accident several weeks before the planned trip, and wasn't ready by the time we needed it for the trip, so we borrowed our old tow vehicle back for that week. (We had sold it to a friend, who lived just across the state line.) No problems, but I did have her write a note, which we kept in the glove compartment, which said it was ok for us to borrow the vehicle, just in case there had been one.
Agreed on spending some time in Wisconsin. They have some of the nicest state parks in the country (IMO), and some of them will have room for your rig. During the week you may be able to get a site evem without an advance reservation. Check into Governor Dodge or Devils Lake SPs - they are very popular, but if your trip isn't for a few weeks, schools will be starting and the parks will be less busy durin the week.
Effigy Mounds is a small national monument/park in northeast Iowa, dealing with ancient Indian culture, interesting place. We spent a few hours there once when camping in s.w. Wisconsin. In central Wisconsin there are not only all the tourist traps in/around Wisconsin Dells, you can visit the International Crane Foundation in Baraboo (dedicated to all the different birds of the crane family), Frank Lloyd Wright's home Taliesin in Spring Green, and The House on the Rock - not like any place you have ever seen before! You could base camp in the Dells or Baraboo (or at Devils Lake SP) and see all of the above (except Effigy Mounds) in short day trips.
Perhaps it's a bit farther from Seattle than you were thinking, but camping in or near Olympic NP might be something to consider for your solo time. Several possibilities: the NP CG in the Hoh rainforest (no hookups), Kalaloch CG along the coast (on a bluff above the ocean), or farther south on the coast, we stayed at a nice little KOA in Bay Center. A short trail takes you to a little ocean cove.
Agree with others who feel this is an awful lot of miles for such a short time. That said, some CGs we have felt were decent (at worst) to very nice (at best) that would be along the two (so far) specific routes people have recommended:
Cheyenne KOA: nice & clean, some sites for bigger rigs, some tent sites, lots of "in-between-size" sites, most pull-through, full-hookup or electric-water. The grounds are almost entirely river rock, which is a bit of a workout to walk in, but isn't dusty or muddy. Road/RR noise not too bad. We have stayed here a couple times & would again.
Rock Springs - again the KOA. Basically a gravel parking lot with hookups, but facilities are clean and it's a fairly easy on-off from I-80. Road noise not too bad, and the redrock bluff behind the CG is scenic.
For the Tetons, you DO have reservations already, don't you?? If not, you will probably not want to pull in too late at night, as you won't likely find a site. Haven't camped at Gros Ventre or Colter Bay (both NPS CGs, mostly no hookups) but many do seem to like them. Colter Bay RV Park if you need hookups - by far the best choice for the Tetons, IMO. BUT you DO need reservations.
Three days for the Tetons isn't actually a bad timeframe to visit; however, since you're right next door, especially if you haven't been to Yellowstone, you should spend one of those days visiting YNP. Spend a full day, drive the south loop in YNP - covers Lake, Old Faithful, Hayden Valley (buffalo), Canyon (waterfalls), and a host of other thermal features, potential wildlife sightings, etc. The other 2 full days you can spend enjoying the Tetons, and take a "side trip" into the town of Jackson. If you are towing a trailer or driving a large motorhome, do NOT try to visit Jackson (the town) with that rig - use your tow vehicle for that.
Fishing Bridge is not privately owned, but it is privately operated, by the park concessioner (Xanterra). Not sure whether the arrangement between GTNP and Colter Bay RV Park is similar, but I suspect so.
About 30-40 miles outside Yellowstone, in the Caribou-Targhee National Forest, there are a few CGs (Buffalo, Upper Coffee Pot and I believe one or two others) that have a few electric sites. Most of those are reservable.