It is just tough giving advise on these subjects simply because I've gone through a similar story earlier on and found out the hard way.
In my opinion, "3/4 ton" trucks or the F250/2500HD line are made more for "bumper pull" type trailers.
The lower GAWR on the 3/4 tons just aren't rated for very much in bed load capacity.
One could easily get a 5th wheel and be under the GCWR but 9 out of 10 times a regular 30'+ long 5'ver will put you over your GAWR and GVWR.
You would have to find one very small and light 5th wheel.
going to an F350 SRW gives you a 1,180 lbs (7,280) more on your GAWR but you still have the 10,000 lb GVWR.
Jumping to a dually gives you a 9,000 lb GAWR and your GVWR gets bumped up to 13,300 lbs.
like I mentioned, I know the pain of dealing with the weight number game and I bought the new truck with the notion that it's job would be safe towing first, normal driving second.
Thousands of folks will tow whatever they want to because physically the F250 will pull whatever you want for the most part. The bad part is that it is not recommended and against the law to tow above the listed numbers and many people take the gamble they will never have an accident or get checked.
Back to the Trailer, most of the smaller Heartlands I've looked through still have a pin weight of 1,700 lbs or more with a dry weight of 10-11,000 lbs while a bumper pull travel trailer at 33' long weighing approx 9,000 lbs had a hitch weight of 870 lbs which would leave you with some left over weight .