Looking for your suggestions

My Dodge 1500 makes a very nice toad, IMO.
Next summer I will be trying it with a Kawasaki Brute Force 750 in the bed.

I think most Chevy GMC products are towable 4 down. But you can check with the annual publication by FMCA or Motorhome Magazine, they both have archives. Naturally you will want to verify this with the owners guide. It sounds like your towing load will exceed the 5000 lb limit on most if not all gas rigs, both the hitch limit and the GVCWR. Just things to think about.
Happy Trails.

OR another way to go would be a MC lift on the back of the MH and tow any toad you want.

If it were me I'd look at a Toyota Tacoma 4 door double cab 4X4 with a Remco driveshaft disconnect. and an SMI toad brake system

I have carried bikes in the shortest of short beds. May need to go diagonal and/or leave the tailgate down.

I would go for a small 4x4 like a Colorado, then you have that as an off-road option in addition to the DR. I carry a DRZ-400 at times myself.

As with any potential toad you need to research the particular make and model. As a very general rule, 2WD automatic pickups are only towable with a driveshaft disconnect. Many, but not all 2WD manual pickups are towable, and many but not all 4WD pickups are towable without modification. As far as hauling the bike, yours is less than 400 lbs so virtually any pickup with handle it easily as long as the bed is long enough. Many 4 door smaller trucks have very short beds that your bike may be too long for.