OP wrote:
How/why does the automatic trans have any bearing on what the trucks suspension/brakes can safely support/stop/tow?
The tranny has noting to do with brake specs.
Brake specs are a function of the GAWRs.
Example is your '06 DRW Dodge/Cummins has a 5200 (4x4) FAWR and a 9350 RAWR = 14550 lbs of braking performance. GVWR has nothing to do with max braking performance on a truck or trailer.
Those big 9350 RAWR on your DRW is the biggest reason the DRW Mega cab is a very popular truck with commercial and private haulers pulling GN/pintle and 5th wheel trailers at 26k-35k GCW and do it safely/legally for hundreds of thousands of miles.
LDT makers realize many folks that purchase a truck with a diesel/manual tranny may not have ever operated combo like that. A newby can damage a drive line quick when carrying max loads if he/she can't make a smooth launch or up and down shift.
And in some cases the clutch its self is the weak link however I see a '06 G56 3.73 gears has 2000 lbs more tow ratings than the 48RE 3.73 gears in the same truck.
The 40' Cyclones has three 6k axles which can make for some light pin weights. Three 6k axles = 18000 lbs of braking performance plus your truck 14500 = 32550 lb of combined braking performance. This is why we see so many one ton DRW trucks pulling some very heavy loaded 40' GN triaxle trailers.
I would check on heartlandowners.org forum and see what other Cyclone owners report on pin and gross weights. Good luck