Tragic for both families - well, all 3, referencing the young man whose body was found in the Black Hills as well. But there are plenty of warnings issued in and at entrance to the national parks; people don't pay heed and sometimes pay the ultimate price. As far as a "surge" goes, there usually seem to be 1 or 2 deaths a year, minimum, in Yellowstone, usually because someone did something dumb or wasn't paying attention. We can't protect people from themselves 100%.
x2 on what Dog Folks said.
We did a "cabin camping" trip post-Thanksgiving one year, from Indiana to Texas & back, using KOA cabins. Had to plan according to which CGs would be still open in late November & early December! It worked out well, but it also convinced us that our decision to trade our popup in for a TT was a good one; all the toting & carrying between truck & cabin reminded us of how we used to do setup and breakdown with the popup. :) Still, a good option - and I know I feel safer in a lot of CGs than I do in a lot of motels!
Quite a trip!
We can vouch for the Natchez Trace being well worth your time. It's a beautiful ride (we did the Tupelo - Nashville section). Didn't use the Trace CGs but found a decent private CG in Tupelo only 1/2 mile or so from the Trace entrance. (The Trace, like an interstate, has only limited entrances/exits, but otherwise is a parklike, 2-lane road with a maximum speed limit of 50 mph, lower in some spots.) There's a visitor center near the Tupelo entrance too, which will help with maps, info, etc.
The thing is, one does not see Yellowstone in a day. You'll miss 90% of the highlights, not to mention the hidden gems, etc., that way. Unless you are into marathon driving and have 3 or 4 drivers who can rotate so you can stay on the road nonstop until you get there, I don't advise this time frame. We used to spend 3 weeks roundtrip to get just to Yellowstone & the Tetons from the Chicago area! And that of course did not include the Grand Canyon, or much in the way of sightseeing on the way there & back.
I believe Jonathan Winters was a suicide as well? (Not sure.) I know Robin really idolized him, and they were really kind of 2 peas in a pod - similar personalities, similar talents. Not necessarily suggesting it was a copycat act, but just - well, not surprising. Sad, but not surprising. We will all miss him.
For Pictured Rocks, we stayed at a CG named (something like) Wagon Wheels - private CG, lots of trees. We had a popup then, so didn't care much about overhanging branches, so I can't address that issue. I have since seen pictures of the Munising city/county campground and it looks wonderful! Lots of folks here seem to recommend it. Both have at least some hookups.
We stayed at the St. Ignace KOA, again with a PUP, and liked it well enough. Straits State Park would also be a good choice - it's in St. Ignace as well. No water hookups to my knowledge but they do have electric; I'd really recommend reservations. We have stayed at Mackinaw Mill Creek CG, which many recommend, but it was huge and we HATED the FHU sites we were in at the time (which was over 25 years ago).
We did a day trip to Sleeping Bear Dunes from the Petoskey KOA, where we stayed. Nice CG but not as wooded as we would have liked.
Canada is very strict on guns. If you want to go, leave yours at home. That said, we made a short foray into Alberta last month, without the camper (left it in a CG by Glacier) and spent an enjoyable day in Waterton Lakes NP, which adjoins Glacier NP. Definitely recommend going, if only for a day trip, though the CG in Waterton Townsite (within the park) was surprisingly decent - older bathhouse buildings but sites seemed reasonably well spaced, and the view - either from the campsite or a short walk from it - was just outstanding, down the valley with the lake in the foreground and mountains around it!
The only thing to really watch for, besides the eternal nuisance of road construction in various places, is going through the town of Jackson, Wyoming. There is a truck route, which I recommend over religiously following the route signs. Just a bit less traffic and way less pedestrian traffic, and it's only a detour of several blocks. The sign is fairly easy to find if you're southbound, which you will be. Good scenic route. 26 from Alpine to Idaho Falls is a gem. :)
If you want to be in the center of the park, Fishing Bridge is your choice. Otherwise, Grizzly is great - and convenient to the park entrance. Really an excellent choice if you plan to spend the bulk of your time touring the park. We've stayed there multiple times. We have also stayed at the older KOA (there are now 2, both west of West Yellowstone by a few miles) - years ago - and at Henry's Lake SP in Island Park, Idaho. Electric-water connections there only, and a dump station. Great scenery there, but it can be windy. If there aren't fire restrictions in place, you can have a campfire at your site. Not much shade, though, and sites aren't private, though they are spaced pretty well.
Cloud Driver is right, going through Idaho is longer but would be, in your case, better than trying to thread through downtown Jackson! (There is a truck route through Jackson in case you do - watch carefully for it. Easier to find southbound than northbound. Definitely preferable when towing the rig over going the regular route thru town.) 26 from Alpine through Swan Valley, Idaho is one of the prettiest drives we've taken (and can be spectacular in the fall); good road, and there's a nice rest area west of Swan Valley - good place to stretch your legs.
Your main climb will be on US 20 on Ashton Hill, just north of Ashton as you head up into Island Park. It's only a few miles. There has been construction on that route this summer; we haven't been there for over a month, so not sure what the road conditions are there now. Dial 511 from home phone or (if you're in that state) from your cellphone and you can get local road conditions with minimal number-crunching - works in the states around here, anyway.
Gas stops: if you do take Cloud Driver's route, fill up in Alpine. There is a gas station there your rig will fit into. Nothing for big rigs after that until you're basically in Idaho Falls. Island Park has a few stations large enough for you - keep an eye out while driving through there. West Yellowstone's gas is A) very expensive and B) very few large gas stations capable of holding your rig (especially with the trailer in tow).
Your second plan is much more do-able. You will be able to guess my gender when I suggest that you build in a little time in some of those private campgrounds to do laundry. Yep, one of life's little necessities that takes anywhere from 3-5 hours out of a camping trip. (It can be done while you working on dinner, or if you want to sleep in shifts. :) )
Remember to MAKE ADVANCE RESERVATIONS in state parks and national parks particularly. People do show up & find campsites, but if you bank on that, you may find yourself being turned away in Yosemite, Yellowstone, etc.
BTW, I know you're aiming for least expensive, but the CG you chose for the Badlands is basically someone's backyard - or so it seemed to us when we drove by. I'd recommend you look instead into staying in the national park CG (they have recently installed electric hookups, and you WILL need air conditioning there) or at the KOA, several miles south, which has shade trees as well as hookups. The Badlands can be "done" by driving the loop, which starts and ends at I-90, and unless you want to go hiking - remember, in summer, this area can be upwards of 100 degrees - you may be content with just driving through. Just a thought. That way you could add an extra night up in the Black Hills (more pleasant) or somewhere along the way across SD. There may be some Corps of Engineers CGs along the Missouri River, mid-state in SD, which could be somewhat inexpensive.
Walmarts can be somewhat few & far between in some of the western states. Not sure how many there are between Yellowstone & the Black Hills (Mt. Rushmore) - Cody has one and I'd guess Buffalo, Wyoming might have one; perhaps also Gillette (WY) or Spearfish (SD). That would be about it. The direct routes between Yellowstone & the Black Hills cross the Bighorn Mts - slow going, and make sure to watch engine temps on the way up and brake temps on the way down (experience talking)!
My beef is that, though the book is even bigger than the old TL directory (which we used regularly), it has even less information about CGs that aren't specifically Good Sam members. State parks are almost nonexistent in there.
Yes, we've towed a large TT across the country, several times. Good advice to plan ahead for spots you want to visit and to buy good RV-oriented roadside assistance (GS has worked well for us, but there are some competitors out there; bear in mind one may work well in most areas but in a few areas, not so much, since they do depend on local resources).
I would add, travel LIGHT. That's especially important since you may be at or near your weight limit. (Is that 6366 figure the trailer's unloaded weight, or is it the GVWR of the trailer? Remember that onboard water & propane count toward the total weight too. You don't want to overload the trailer, so take less than you think you need. You may have to do laundry more often, but that's ok. (Save your quarters ahead of time, if you can.)
More advice: when laundry time is due, stay at a private campground whose ratings mention clean laundry room. Worth it, trust me. A lot of state parks will say they have laundry facilities, but most of those will be a communal utility sink. :)
Not a ton of choice in the SLC metro area. We have stayed at both the SLC and Brigham City KOAs and they were decent, though nothing to write home about. SLC's advantage is its convenience - even more so now that the Trax commuter train line is finished on North Temple St. We were last at that KOA when they were building the line, and it was a bit of a hassle. We did find the side road out, though, and that turned out to be a little easier access. We did enjoy the country atmosphere at the Brigham campground too.