My 3.5k rated minivan had the same engine, gearing, and brakes, while being a little heavier, longer wheelbase, and higher payload capable than the 3-row CUV built on the same platform. The FWD CUV variant had the same 3.5k rating, and the AWD CUV variant got bumped up to a 5k rating. I believe this is pretty common across manufacturers. We towed a 3.5k travel trailer with it (and an Andersen WDH) for a few years, and while stability was fine, it was drivetrain limited. Based on my experience, I would definitely recommend against pushing the tow rating with anything (including trucklets based on the same) in this 'barely 5k' rated vehicle class, especially in more hilly/mountainous areas.
I had long loathed the idea of full-size trucks. However, while I like the idea of mid-size trucks, we immediately found during our search for a more capable tow vehicle that the reality of them wouldn't work for us. While certainly not great, since I don't actually spend a lot of time in aggressively tight surroundings, the girth actually hasn't been as bad as I had feared. Even the fuel economy has been shockingly good, especially since it somehow manages to relax my driving style (and me) in a way I could never achieve in more 'fun' cars. In a very different way that we typically talk about around here, it very much doesn't suck to drive for me. Honestly, and as surprising to me as anybody, my near-ideally (unicorn) optioned 2.7EB 10-spd F150 is one of my favorite vehicles I've ever owned.
For regularly needing a 5k tow rating, I'd say at least a 6k+ (or better 7k+) rated vehicle with RWD (rather than FWD) based drivetrain. The pros/cons of of a truck bed vs a SUV hatch are situationally dependent. Both are compromised, it's just which set of compromises ticks the most important boxes. For us that meant a truck bed. Admittedly, if I was in a different situation that didn't need as much rear seat room (2 kids + medium dog) I might have gone with a (high payload rated) midsize truck as being more 'right-sized'.
Long story short, from what little I know of your situation, I'd probably be looking at mid-size trucks and/or similar size and capability C/SUV's... But keeping a close eye on payload.
Without actual measures, I'd conservatively protect for up to ~700lb tongue on your trailer. Depending how much more than just the car you bring with, I don't think I'd want less than a 1300 pound (drivers door sticker, not over-stated advertised) payload rating in this case.