I’ve never made a single 10,000 mile trip, but I made a lot of them over the years that were 4,000 miles plus. But my PU was an older all metal Coleman, smaller and a whole lot lighter than what you’re proposing. In fact, the newer decked out PU’s I’ve seen are heavier than my present 16’ TT.
But it worked fine on all our trips, no complaints at all. And the light weight and low profile hardly dented my gas mileage compared to the TT. I can’t say it was a tiring drive; it was actually an east pull, but I suppose it depends on how long one drives. The “weak” point with mine was the smaller tires. I found I was probably going to lose at least one tire on a trip and I’d sometimes lose a wheel bearing. So I carried and extra spare tire and wheel bearing. I got pretty adept at replacing either one along the road or in a parking lot.:) But the bigger ones have larger tires so that may not be an issue to you.
Make sure your TV is ready for a long trip. Check the tires on the PU and if they’re older or cracked replace them before you go. And grease the bearing. I didn’t take my home tool chest with me, but I did take tools that I would need to work on the electrical or plumbing systems. With either a TT or PU most problems you’ll likely have will be an easy fix that you can do yourself.
Both the PU and the TT have pluses and minuses and it’s your call which you want to use. The PU will give you better mileage and is probably an easier tow even given the size, and the TT is more convenient. I don’t know how having dogs in a PU will work, I never did that. But it sounds like you have a really great trip planned, so good luck.