No experience with them, but I did find this... Toyota says you can tow up to 5000# with the FJ....

http://www.toyota.com/fjcruiser/specs.html

http://www.toyota.com/pdfs/towguide_Part1.pdf

http://www.toyota.com/pdfs/towguide_Part2.pdf

http://www.toyota.com/pdfs/towguide_Part3.pdf

Most trailers these days have trailer brakes, but the question I always ask myself is "Can my tow vehicle stop this trailer by itself safely with no help from the trailer brakes?". I would rather be overfaced with my tow vehicle than underfaced.   

I would also be concerned about sway from the short well base of the FJ Cruiser.

I have no experience with the FJ, but the specs put it in the same range as the Trailblazer.  I would give it a try, there is no reason it couldn't haul a small horse trailer.

As for the newbs crying about the wheelbase and brakes; the overhang on that vehicle is almost 'nada........but I guess you don't fully understand leverage to begin with.  I couldn't find the size of the brakes listed anywhere, but the 60-0 distance was listed at 127 feet.  That is a full 30 feet shorter than my Duramax, this ADVANTAGE does not magically disappear when hooked to a trailer.  More weight takes longer to stop, that is basic physics, I would put money on the fact my Trailblazer would stop faster than my Duramax pulling the same trailer.  I won't bother to address 'losing the trailer brakes', that is a bad day no matter the tow vehicle.

To begin with the OP asked about a "short" 2H, to me that means no DR.  A small 2H can easily come in under 5K with one or two smallish horses.  Secondly if your 2H comes in at 7K that is one heavy trailer, my trailer is steel, 15' on the floor and 6'8" wide, coming in at 6K loaded.....I actually consider it larger in the 2H variety. 

As for the wheelbase/overhang, I fail to see how you think the lateral forces will rotate around the COG?  A lateral push on the ball, has to rotate through a physical pivot point.  Meaning the front axle is going to pivot around the rear axle.....unless you push hard enough that the rear axle loses traction, even then it will pivot on the front axle not the COG.  I suppose one could argue that if the COG was exactly half way between the ball and front axle it may be rotating at the COG, but even then that would assume both physical pivot points would have to be compromised and we would be well on our way to a wreck at that point........

If you think out of two vehicles with roughly the same size contact patch, one weighing ~2000 lbs more than the other, that the heavier one will stop sooner.......well I don't know what to tell you.  Downforce is exactly the reason the smaller FJ can stop 30 feet quicker than my Dmax.  On tires that are more or less the same size, the additional "downforce" of the heavier truck over comes the avaliable traction and the clamping force at the brake calipers must be reduced to avoid skidding.  Even with the additional 500#, the lighter truck still should have more available traction for stopping.

Additionally since a WDH will be required on the smaller vehicle it may hold even further advantage.  With the WDH during a hard stop, the weight the trailer transfers to the tow vehicle will be evenly distributed to both the truck's axles as opposed to a WC setup where the rear axle is more heavily loaded as the front is proportionally unloaded.....causing loss of braking and steering.

Considering your statement, "I'm not out to cast doom and gloom on using smaller tow vehicles. I used an older 1/2 ton for several years." it would appear you somehow consider a 1/2 ton "smaller".  Since I am aware that physically a 1/2 ton can be every bit as long/large/heavy as a 3/4 ton, I suspect we will be unable to have a successful discussion.

Way back when I was young and not so smart I towed A LOT with a Jeep CJ-7. Similar wheelbase to the Toyota.    The handling characteristics were poor to say the least.  I know that the Toyota is many years newer, better brakes ETC.  BUT I contend that the wheelbase (being similar) is a limiting factor as far as safety and control are concerned.  IMHO

How much do you like your horses??

Every Sunday, there is a guy up at the trail head with the rig pictured below.  It's 2 wheel drive Ford Ranger pulling a bumper pull trailer.  He seems to get there safely each week.  He mostly arrives with  1 horse and 2 dogs in the trailer.  He lives close by and never tows very far.  All of his driving is in the city.  So no hiway driving at higher speeds.

You will need to evaluate where you are going to tow, how much work the tow vehicle will be doing. How dependant you are on actually being at a destination at a certain time.  do you have the luxury of saying,  It's windy today, I'll stay an extra day and go home tomorrow the conditions are better. Rough roads, passing semis, strong cross winds will all effect how well any tow rig behaves.

Yes a FJ will tow a 5000 lb trailer.  And I wouldn't hestitate to tow a trailer full of tree branches to the local dump or a wave runner to the nearest lake.  But I wouldn't hook up and tow a maximum load multiple hours up a twisty mountain road that might have strong cross winds and steep grades.

Another consideration is the wear and tear on the FJ. They are not designed to be a beast of burden for towing. Repeated heavy towing will most likely cause some premeture failures of some parts.  A neghbor of mine recently sold his Tundra after spending a bunch of money repairing it.  He had used chains on his front tires during a hunting trip because of the mud and snow.  the mechanic informed him that those vehicles, while being a 4x4 are not built for the added stress that the increase traction from chained up tires provided. The FJ may be able to tow the trailer, but frequent towing my result in increased maintenance for you.