Hi Holger, Welcome to Expo.
Obviously this is my opinion only........... I just bought a Strom and there is no way I'd say stick that out the back of a camper.Way too big. You want some thing as light as possible. A Madass would be cool and super light but if you want a dirtbike then that wouldn't cut it either.
I also reckon you need about 250cc if you intend to double and if it's only for relatively short hops to the shop or to duck down some rainforest track then that should be enough. No you aren't going to be comfortable on the highway but as a support bike that meets the criteria you described, I think that size would be fine.
Ag bikes are a great option. Yamaha do a good AG 200 and you'll find parts and service for them in every outback town with more than 2 pubs. Just not sure if you can get them with a double seat.......hmmm. No rocketship but very compact and ultra practical. Come with stuff like big mudflaps, crash bars, very comprehensive toolkit, enclosed chain, front and rear racks, knobbies, handbrake, oversize side stand, etc. ...extra low seat and overall height too which should make them easier to load on to a high rack ......well, I presume your rack will be extra high if it's going onto the back of an expedition-style FG. BTW PM me if you need a winch arm to drag your bike up to a high rack. I know a guy that makes racks that clip into a 2" reciever and they have a little winch arm that is removable....so it sort of goes in the end of the rack and you use this electric winch with a pocket remote control to drag your bike up. Then remove and stow the winch arm and you're away. Makes loading a bike up high a much safer proposition, especially if you are on your own.
The KL 250 or Sherpa below is probably the pick of the Ag bikes although you could almost call that a proper trailbike. These things are relatively cheap but all the Japanese makers do good little bikes like this too so have a look around at what's on offer.
If you are on a super tight budget and still want something brand new then these fit the bill....they are a Yamaha TW 200 knock off.....called a Laro V Retro 250. I don't know too much about them but as genuine TW's are about as rare as honest politicians here in Australia...then they might be worth a look. Perfect for a little trip up the beach too........ and that seat looks like it would be a bit more comfortable for your missus than your average pillion seat on a conventional 250 trailie.
We made a hydraulic bike lift for one of our Fuso motorhomes a couple of years back and the owner imported a Rokon Trailbreaker to hang off it. Crazy bike but definitely suited the truck. That whole rack with the spare wheels, Rokon (also another mini pitbike which isn't shown) and the outboard comes down to the ground with the push of a button . Cool huh?
Good luck mate.
Regards John
BTW where are you based????