GeoBoy, first, you need to determine what you want to do with a scooter and this will determine the size of scooter that will fill your predicted needs and riding expectations. It sounds like the two of you will be on the scooter at the same time. That generally rules out anything below a 150cc scooter.
You also mention touring ... that could mean an hour away or a full day's journey round trip. If that's the case, you won't always have the luxury of using secondary or tertiary roads and you'll want something that will be comfortable for both of you over that distance. At minimum, you'll need a 250cc scooter that is capable of carrying both of you at say 60 mph.
We tow a single axle (rated at 3,500 lbs. with electric brakes) 6x10 enclosed cargo trailer behind us in our truck camper set-up. We have two larger maxi-scooters in it (a 600cc Honda Silverwing and a 400cc Suzuki Burgman) as we both ride. The cargo trailer also carries all of our camping gear (6 chairs, large propane camp stove, full tool box and accessory boxes, all motorcycle riding gear, oil and related stuff, mosquito net gazebo, even firewood etc. As you can imagine, we pack it pretty full and the total weight of the trailer (around 1400 lbs. empty) is about 3,200 lbs. fully loaded (weighed at a weigh station). The two scoots alone are 600 and 500 lbs. respectively.
We're considering adding a roof rack to transport a canoe and plastic kayak. That would put us pretty close to the GVWR for this trailer. Right now, we do not worry about what and how much gear we put in the trailer ... it can handle anything we throw at it because we know we're below the maximum weight rating. In our jurisdiction, if you go with a tandem trailer, it is subject to annual safety inspection and licensing fees ... the 3,500 lbs. class, single axle trailers are not.
Given what you described and to answer your question, I would look at single axle (you truly do not need a tandem axle set-up), with good D rated tires (C would probably do depending on how much weight you carry), rear ramp door access trailer, with electric brakes that is rated at 3,500 lbs. (a standard rating for these trailers). A side "man door" is also nice for easier day-to-day access to grab things you've schlepped along.
My recommended specs are a 6x10 or 6x12 V-nose (mine is a bull nose front but I bought it used). It should be tall enough to stand up in (mine is about 3" short for me ... again I bought it used so had no choice ... my bruised scalp wished I hadn't). You can have aluminum racks added to the roof to carry your watercraft and if it were me, I'd add a passenger side awning running the full length of the trailer. This is good for an additional seating/snoozing/working area (for scooter maintenance etc.) and ideal for parking your scooter under in the bright sun or on rainy days.
Hope this helps.
2007 Adventurer 90fws Truck Camper
2001 FORD F250 SuperCab; 8' box; 4x4, 7.3l diesel, rear Sumo Springs