I haven't had any overheating issues at all. I run a Scangauge II so I can keep an eye on fresh-water temperature and transmission fluid temperature. I had also heard/read rumors of overheating, Ford warranty voiding, etc. But I found some solid info that made me feel like trying it (it's easily reversible). A fellow with a Born Free (very similar to our Chinooks, size, chassis, and engine wise) had done a long trip to Central America. He'd mounted the front spare and also ran a Scan Gauge. No issues. If it could get him and his family/gear through Central America, I figured I'd try it too.
Around the same time (but a month or so after the tire) I put on a larger transmission cooler. This also hasn't changed the fresh-water (engine cooling) temperature but I think it does cause the cooling fan to come on slightly more often to maintain the temperature (typical FWT for my rig is around 198ºF). Say, when chugging up a pass in hot weather. The reason I'm pretty sure it's the transmission cooler and not the tire is that the tire is quite a ways out in front of the grille, whereas the transmission cooler is right in front of the radiator (as the stock one also was) and of course it not only blocks the radiator slightly but it also spews heat (as it cools the transmission). If I were to change anything, I'd probably see about fitting a transmission cooler that was sized about halfway between the original and the one I added. On the other hand the engine cooling does have the fan, and when it comes on it cools the engine down in about ten seconds. (And it did come on even before the tire/trans cooler too of course.) Whereas the transmission doesn't have that ability.
I have done suspension mods. When I got my rig, it was pretty much bone stock and had 49,000 miles on it. It has a leaf spring rear suspension (no More-Ryde). First thing I did (suspension wise) was put on new shocks. I went with Koni FSD. Then I drove it a while. Nothing was horrible, but I like something fun to drive, so I still wanted to make some improvements.
Next thing was a new (2014) front end (take-off from a van being converted to 4WD). This was a HUGE improvement in braking (that's the reason I did it), but also gave me the somewhat improved newer front suspension (2008-onwards), and of course also resulted in all new ball joints, etc. This was such a great mod that I'd do it again in a heartbeat. No more dreading the "stale" green light. (My original brakes were in good condition; just seemed under-braked. And the new brakes and front suspension are MUCH beefier than the originals.)
At the same time I put on a Roadmaster front and rear anti-sway bar. This was great for handling, but caused the rear end to get harsh (when going over bumps). Not acceptable. I did some experimenting (helped by the good folks at Henderson's Line-Up in Oregon) and ended up changing to a stock E-450 rear anti-sway bar (is sized between the stock bar and the Roadmaster bar), plus going to a Koni adjustable shock on the rear (adjusted to softest setting). This gave me the goodness of a beefier sway bar, but without the way-too-harshness of the Roadmaster rear bar. (I kept the Roadmaster front bar.)
I also added a Safe-T-Steer to the front, although again I found the "recommended" size to be a bit much. Great on the highway but made a twisty road or driving around town a bit too much of a chore. So I went to a smaller version (I think made for a Toyota Dolphin). That got everything to be just about right for me. One note is that Henderson's has a "test route" and when you arrive there they take your rig (and you) on a long drive around this route (many miles) and then give you a score sheet showing everything you could/need to do and what all the handling faults are. They said the Chinook was already pretty darned good (I did already have the Koni FSD shocks, the front Roadmaster bar [had the stock E-350 rear bar on for the test drive], and the 2014 front end). But I did at that point still elect to do the rear bar and shock tweaking and the Safe-T-Steer installation.
Other options I considered (and would still consider) for the rear are Sulastic shackles, More-Ryde, parabolic springs, and air suspension (in order of least $ to most $). I'd likely try the Sulastic shackles before anything else. The goal would be to maintain the fun but reduce rear harshness (even more). Perhaps then the rear Roadmaster bar could be used.
I still don't mistake the Chinook for a Miata, but it's pretty fun to drive. (And a Miata lacks a kitchen, couch, and bathroom
).
If my only option was putting my spare on the roof i just wouldnt carry one. Spares not much use when you break your neck trying to get it down.... 
