[wingnut60]

If you are going to stick firmly with Ford's tow ratings, then your figures are correct. You should have gotten a DRW.
However, there are plenty of 250s towing such fivers, so it can be done, maybe not wisely.
It is up to you to decide. You can't change the ratings no matter what options you add, but you can make it handle the load better.
The MAIN factor in my opinion is the tire capacity--I would NEVER exceed the rated capacity on the rear tires--and that is what is going to cause the main problem. You probably have no more than 3500lbs/tire, leaving you 3180 capacity for passengers, etc., and pin weight. Since you have the truck, do some hard research on possible fivers with lightweight ratings.
If tire capacity is the main setback, you could consider swapping to a set of 19.5" wheels/tires--about a $3000 upgrade. Going over the RAWR would not bother me if the tires were not overloaded.
On the other hand, you might come close to not much more than $3k for a '14 dually....
Look at fiver combos as you travel around--you will see lots of 250/350s towing larger fivers--maybe not right, but done often.

On the other hand, since this is a long-term decision and for full-timing you would not want to compromise on the fiver--maybe you should reconsider upsizing the truck at a loss so you can consider the fiver you both would really want to live in.
Another consideration--if you paid more than invoice for the truck, you were at the wrong dealer. There are several internet sources that will sell at invoice, then take the incentives off.
Sorry that you are in this situation, but almost everyone in RVs has done something similar at one time or another--myself included--and just have to recover somehow.
Best of luck,
Joe