6x10 Enclosed utility trailer-safe w/no brakes?

I'm assuming you are looking at something like this trailer (link)

Shouldn't be a problem with it loaded properly. That trailer for example is 1,100lbs empty with a max GVWR of 2,990, either way it should be fine. Just remember to down shift when going down large hills if needed to keep your speed from increasing. Use your brakes as little as possible but when needed. Your engine will rev more than normal when going down hill when down shifted but don't worry unless it's getting over 4K RPMS then you may want to hit the brakes to help slow you down. Don't tow in OD or 6th either.

FWIW, I just pulled a larger 3,200lbs trailer without trailer brakes (or weight distribution hitch) for 3 days through the Rocky Mountains. I have a manual transmission and it easily kept my speed down while going down large hills (note the trailer did have more wind drag than a cargo trailer). Remember that braking distance will be increased with more weight so leave plenty of room in front of you and other cars. My stopping distance was probably about 1.5 times of normal. Not too bad but if you weren't paying attention it could get you when coming to a complete stop.

Also, you probably won't be going over 70-75 even on the Interstate with 3K behind you. Just go slow when needed and be safe. You'll get the hang of it and figure it out quickly.

You didn't really say how much cargo weight will you have in the trailer and inside the FJ. I think you could creep into an unsafe zone if you're going near max capacities. And conversely if you're super lightly loaded you're likely in the way safe zone. Also you didn't mention your experience with towing. I apologize if this is mostly info you already know.

I too would add a transmission cooler before I towed over about 1500 lbs up mountains with an FJ, especially if I was going to be doing it on a regular basis.

Regarding your vehicle brakes, overheating them is really your worry. This is why others are suggesting you should avoid using the brakes and instead shift down to utilize the engine slow you down. You want them to be cold when you do actually need them.

Also there are different laws about trailer brakes that you might want to consider (like Nevada and Mississippi): Trailer Brakes - AAA Digest of Motor Laws
I wouldn't worry too much about them, but I'd like to know if I'm doing something out of compliance.

Sway is the other thing that trailer brakes can help with (assuming we're talking about brakes you can operate from inside the cab; i.e. not surge brakes). If the trailer starts swaying, you can activate the trailer brakes without touching the vehicle brakes and it will calm everything back down. You can mitigate sway other ways such as with proper loading (heavier items towards the tongue of the trailer, lighter or no items behind trailer axle) and with sway control devices. If you have awkward cargo that you can't load correctly, then that might be another reason for brakes or adding some sway control equipment.

Another thing most people don't realize about trailer tires is that they are not speed rated to go on the interstate at 75+ mph. Most of them are speed rated 55-65 mph. Radial trailer tires are rated higher (and are much more expensive). If radials are swapped on a trailer designed for bias ply, the ride quality of the trailer could suffer since radials don't absorb the bumps like bias plys. I have changed a lot more trailer tires than vehicle tires in my lifetime and I only tow rarely--I'm convinced it's due to going 70mph.

I could be getting into real overkill area with this part, but if you are going super heavy inside the cargo area of the vehicle and your only option is to go very tongue heavy on the trailer, a weight distribution system might be a consideration.

And I think more important than all of those things is to take your time when you tow a trailer.