herbert007,
You might be OK, though it'll take some work on your part to verify this, and then have faith in a good frame shop to fit your rig with the proper hitch and have it properly welded to the motorhome frame.
Here's how it might work for you. Your motorhome is 31' long. Have you ever had it weighed, loaded as you travel? If not, that's step #1. The GCWR of the chassis is most likely 26,000 pounds. If your rig, loaded for travel, weighs in at about 19,300 pounds (26,000 lbs. - 6700 lbs.) or less, you could tow your truck and be "safe" by the numbers. Assuming we have the weight of the truck correct. It probably wouldn't hurt to weigh the truck as well.
This will still require you to find a frame shop to put a better hitch on it rated for this extra weight and have it properly welded to the motorhome frame.
As others have mentioned, you WILL need a braking system for the truck as well. The motorhome brakes are NOT rated to handle the GCWR of the vehicle, from what I remember reading from Ford literature.
If your rig weighs in about 20,000 lbs. ready to travel, perhaps you can lighten that a bit by travelling with minimal water and possibly other things. If you're a couple of hundred pounds over, it probably won't be a huge deal.
You said you wouldn't be travelling far. This doesn't really come into the "safety" part of the equation, but it does figure into the overall wear & tear on the rig. Especially if you're over the chassis weight ratings.
For comparison, our 36' gas rig on a Workhorse chassis weighs in about 19,000 lbs. empty (no water, no supplies, driver only). So it's possible that maybe your 31' rig might be 1,000 lbs. (or more) lighter. (I think I remember seeing somewhere that the Workhorse chassis's tended to be heavier than the corresponding Ford chassis in that era, but not 100% sure of that.) Anyway, not sure by how much, but yours SHOULD be lighter. (Our rig has 2 slides by the way. If yours only has one, that'll be a couple of hundred pounds of weight saved right there.) Fully loaded and ready to travel (we are a family of 5), we're usually at just around 21,500 pounds on a chassis with a GVWR of 22,000 lbs. We tend to pack heavy, full (75 gallons) of water, and I think my wife even packed the snow parkas when we went to Florida IN THE SUMMER!!!! LOL Anyway, so we're just under the GVWR. Which we need. The GCWR of our chassis is 26,000 lbs. and we tow our Honda Odyssey, which weighs in around 4,450 pounds, give or take. Combined we are just at the 26,000 lb. GCWR, give or take a couple of hundred pounds or so, and the rig handles it pretty well. BUT...we haven't attempted the mountains out West. But no major issues with anything on the East Coast.
Our rig had a 5,000 lb. rated hitch on it from the factory, so we didn't have to have any hitch work done. We DO have a braking system installed on the Odyssey, and I can TELL if it's not turned on. There is definitely a noticeable difference in stopping power with and without the TOAD brake system engaged.
Well, as others have pointed out and you said, you have a bit of work to do. I say you just might be able to do it AND stay within the chassis manufacturer's ratings. It'll require a better hitch, but you'll still be by the numbers. Now, if you go OVER the numbers a bit, then it's on you as to consider "how much" is "too much". You shouldn't go over the ratings, but I'm sure there are quite a few who do, and most don't even realize it.
Good Luck, and Merry Christmas to you and yours!!!
~Rick
2005 Georgie Boy Cruise Master 3625 DS on a Workhorse W-22
Rick, Gail, 1 girl (27-Angel since 2008), 1 girl (22), 2 boys (23 & 20).
2001 Honda Odyssey, Demco Aluminator tow bar & tow plate, SMI Silent Partner brake controller.