I have moved a lot due to my profession and learned a long time ago to lie to Uhaul. According to them my car will not fit on their trailer (front will hit) so they will not rent it to me, I always told them I was hauling an 88 Yugo GVX.

 

I hauled a old massy tractor inside my enclosed trailer 2 weeks ago, load was probably around 6000lbs. It towed just fine.

 

I have a 4.7, and they rented to me when I needed to go get my grandma's 98 buick lesabre, and that must weigh more than the 'Stang. And as far as capacity of the factory hitch, If I haven't bent mine yet, no one else on here ever will.

 

Pulled my Mustang several times using a Uhaul trailer with my 2008 QC Hemi. Pulls just fine. I would tell them I'm hauling a honda civic because I wasn't sure what their weight requirements were. I used to have a 2000 Silverado and they denied me trailer rental for my mustang because the truck didn't weigh enough(wasn't extended cab) which sounds ridiculous. So, that's where I had they idea to tell them I was moving a civic and they allowed me to use the rental. NONE of their business what your towing IMO. it's your insurance not theirs. Just like they always want to hook up the trailer ball/wiring&chains to your truck(I tell them don't touch my truck unless they want to be liable if it comes off down the road...which they won't be cuz it's on you as the driver)
Ram tow hitch on a 2006 is a Class 4 not a 3, your fine. I've towed plenty of loads heavier then my 5k rated reciever as well.
As for brakes, Uhaul trailers use Surge brakes so you dont need a brake controller for the trailer brakes.

 

I am a U-Haul dealer, and I can say your truck will have no problem towing a Mustang on an auto transport.

Those of you who want to lie when renting, go for it. It invalidates the whole contract meaning they're on the hook for nothing if anything goes wrong. God forbid they want to be sure you have a safe tow experience and don't want your vehicle to be damaged! They've only rented trailers for 65+ years, mostly to people who have never towed anything in their lives. Their stupid safety rules are in the way.

As far as who hooks up the equipment, people bitch when you don't do it too. "I rented a trailer and they just said here's the paperwork, have a nice day and left the rest to me." Many people who pull boats & utility trailers don't know how U-Haul's hitch or chain setup works. Again, how dare they do it for you!! Damn that customer service! When the dealer doesn't hook up the trailer, guess who gets blamed when it does come off -- that's right, not the customer!

You can play "what if" at uhaul.com to see if you can tow the vehicle you want with your truck. Just pick the auto transport, fill in your truck specs, and see what it says.

 

Now see, I tried the what if at uhaul.com couple years ago and it told me I couldn't tow an 88 Nissan Pulsar with my truck, which is insanely laughable since I can just about dead lift the back end of a Pulsar, so I don't know how accurate that thing is.

 

I have been dragging around 6 and 7 thousand plus pound rv's with mine since it was new, with no problems. Of course those are made to be towed and i always used an equalizer hitch. But if you are using a trailer like the ine pictured, with dual axles, should have no problem.

 

Bitter, party of one, your tables ready!

 

Then I suggest you to try some one other then Uhaul.
There are lots of companies which can ship vehicles try them.

 

The U-Hual pic above appears to show two 3500 pound axles, and if we take into account the 10% - 15% tongue-weight standard, it should be able to gross at about 8000 pounds - the trailer prolly tares @ 2000 so you'd have 6000 pounds of cargo capacity when connected properly.