I'm not sure why you would have the Tow/Haul button with no tow package, but, I'm going to guess that you could. The Tow/Haul button alters the transmission settings only. With the Hemi, it also appears to disable the MDS system. Since it is transmission related, it is possible that it would be a "standard feature" since it is Tow / Haul
and as such would be useful for hauling a heavy internal load.
I would think, even if the previous owner removed the hitch (I couldn't imagine why), he wouldn't have removed the wiring harness...you should see remains of it around the spare tire if he just cut the ends off.
My guess is you don't have the Tow Package which means that you wouldn't have the wiring harness, relays, cooler or upped alternator. Without the tow package, you'll also not have the relays or the trailer wiring harness installed. All could be done by a DIY, but, I believe you would still need to have the dealer let the vehicle know you have the tow package with the StarScan tool. If the Commander's brain doesn't know you've installed the relays and the wiring harness, it won't know to pass the charging circuit and brake light pulses down the trailer wiring harness to the trailer connector...
If you get the Tow Package installed by the dealer or if you do it yourself, they do make an additional harness that you can add that would put a brake controller plug under the dash like you mentioned - I had the tow package, but, for some reason Jeep doesn't consider the brake controller wiring a part of the trailer wiring harness...the wires are there and I easily added it, but, I was a bit annoyed that it wasn't part of the overall trailer wiring harness; seemed silly to me.
I tow a 23' FunFinder at around 6300 lbs ready to roll. The Jeep has two concerns the 350 doesn't...shorter wheelbase and the full time 4 wheel drive. On both accounts, you'll want to keep the weight transferred between the axles for safety and longevity of the transfer cases....you'll want a good Weight Distribution hitch (Reese Dual Cam is good, I use Equal-i-zer) and good sway control. You'll also have to watch your trailer's tongue angle. You'll want to keep it level or slight down angle...that much weight you don't want lifting the rear end up in a panic stop should you have a problem with the trailer brakes.
My setup: 10K Lb. Equalizer Hitch, Tekonsha P3 Controller, 3/4" down angle on trailer tongue.
Can only tell it is back there when starting off the line or steep hills (Tow / Haul helps there).