I have a similar setup however, there are ALOT of extras on my truck. I didn't have the means or ability to own two trucks and my company provides me with a 1/2 Ton so that's what I have to work with. I have a 2011 Ram 1500, 5.7 with 392 ratio, legally I am allowed 10,500. I tow a GN trailer with it often and I tow everything from a 26' enclosed to a 16+4' GN deckover. However, ALL my trailers are liscensed at 9,990 to avoid combination plates and I never haul anything that would put me over 9k.
Yes the truck should have more than enough ummph to pull the trailer. However, as mentioned suspension will be a concern. Get a set of good airbags and this will help solve that trouble. Also good tires on the truck and brakes are crucial. On top of that I have updated the brakes to electric/hydraulic on all my trailers for extra braking.
Finally, you have to watch the weight 3200 unladen on a goosneck, maybe if its alumninum. But I highly doubt that its a correct weight.
My 16+4 Appalachian 10K Gn weighs in at 4,000 empty from the factory. I have changed the floor out to expanded aluminum from a wooden deck and I am down to 3700#. So then you add on a 6000# tractor, chains and binders and you are over 10,000#.
I do not know about the 04, but the new trucks do have a pretty high rating depending on factory options you can see anything from 6800# to 11,200# again depends on options and manufacturer. So assuming you have at least 10k you could possibly be legal. But in PA over 10k and you now need combination plates.
So all in all, you might be better to find A) a bigger truck. or B) if that's not an option perhaps a tag trailer with sway bars would work since the trailer would have a light unladen weight and let you stay legal.
This is all just one man's opinion and ultimately it comes down to what is safe, legal, and what you can feel comfortable with.