Let's look at this like this...
First off, the idea of taking of in a motorhome, towing a flying machine, so you can get some flying in as you go from area to area is a awesome idea and sounds like good times to me. Im actually very jealous!
But you need to decide on what type of aircraft you.... (1) WANT to use for this mission and (2) will said aircraft be suitable for the mission.
You could most easily pack a backpack powered paraglider and fly that. Would be able to be put in one of your motorhomes storage bins, would be cheapest thing to buy and would not require a airport to fly from. But you have to be in good physical condition to run with that on your back up to takeoff speed and you won't want to fly in windy conditions in one. You could go with one that has a trike framework where you just sit and steer rather than run to get airborne, but you still have the wind limitation along with a 25-35 mph top speed so your not flying far from where you took off from in any parachute type aircraft.
You could go with a Trike, which would fit in a relatively small enclosed trailer, wing folds up and can be put in a large tube on the roof of the trailer on inside the trailer if the trailer is long enough ( may want a longer trailer so you can bring more toys along, say a pair of Honda Monkey motorcycles and some kayaks so you have other toys to play with on your stops ) Trike would fly in light to moderate wind conditions, would be preferable to use a runway, but trikes typically takeoff super short and land fairly short as well. Very simple machines and would likely handle being trailered around quite well. These will fly in the 45 + mph speed range, with some models being over 75 mph, so you could extend your flights a good bit away from where you took off.
You could do a folding wing light sport / ultralight-ish airplane. Something like a Kolb, or Kitfox. Wings fold back quickly and easily and stay on the airframe. Some designs out there can be towed on their own wheels such as the kitfox, although I wouldn't tow one like that very far. Same as the trike, they don't really require a airport, as both planes will be in the air in less than 500 feet and a good pilot could land them in the same distance... But would probably be best to use a airport when possible. I would think out west, in some areas it may be easy to find a large open field suitable to fly off, where as in some areas in the east it may be much harder. But there are airports everywhere.
Then there is Gyros... Gyros will only take up a little more space in a trailer than a trike. They typically need more room than the airplanes and trikes to take off in but can land in much smaller spaces. NO gyro really likes taking off from a rough field... some do better at it than others... but all gyros prefer a smooth surface to takeoff from. Like above, you MIGHT be able to find a plot of land suitable to fly off, but you will more than likely be best suited to just find the closest airstrip and go there to put together the rotors and fly. Gyros typically fly at 45-80 mph, with some of the expensive " Euro " two seaters able to fly at close to 100 mph. Basically same speed range as the airplanes above, and same speed range as some of the trikes. You CAN trailer with the rotor on, but that is very hard on the rotor, the rotorhead, the bearings, the controls, etc.... to do it occasionally here and there, not a big deal. To do it across the 48 states, not a good idea. Good thing is, the rotor comes off with one bolt and you can either use muscle power or a winch and get it down off the gyro and either placed on a rack on or inside the trailer, or set on sawhorses and disassembled completely and boxed up for transport. Completely taking the blades off and boxed means a 30 minute process to unload gyro, assemble blades, install blades, pre flight the gyro and be ready to fly... Shorter if the blades are left assembled on a rack, maybe 20 minutes. Time could be shorter if you hurry, but best to take your time and do a really good preflight inspection each time.
The time to prepare each type of aircraft I listed, to fly... and to pack back up once done... is very close to the same. They could all be done in under 30 minutes.
Advantage of the gyro is the ability to fly along and really slow your speed down without fear of a stall.... And the ability to land in a very small space in the event of a emergency ( say a engine out for example )... and the gyro will handle turbulence and winds better than anything else.
A disadvantage of the gyro is you will have to find yourself training... they are expensive to buy ( compared to a nice used trike or kitfox / kolb ) ... insurance will be harder to get and more expensive... And personally, I feel like a gyro is probably the most fragile of the 3 types listed for being trailered around the country.
One big advantage of the trike and or airplanes, is the ability to install and use a BRS parachute system. no parachute system has ever been successfully tested and approved for gyro use. In theory, a gyro is the safest aircraft out there since it can't stall... And trikes and airplanes are well proven to be safe aircraft as well. But in any of these aircraft, failure of just ONE bolt can send any of these out of control and result could be fatal. Gyros typically can not be controlled, if the controls fail. Most gyros, the control system from the stick in your hand all the way to the rotor, there are 8 to over 20 bolts that failure of just one of them your completely out of control. It would be like losing elevator and aileron control in a airplane... except most gyros are never trimmed out perfectly for hands off flight, usually you let go of the stick and the nose drops, and or the gyro rolls to one side or the other cause of the weight of the pre rotator. Of all the safety advantages a gyro offers, the ability to use a ballistic emergency parachute makes some of the other aircraft just as safe or even safer in my opinion.
No doubt though, gyros look the coolest, and probably attract the most attention. And if its windy, the gyro will handle windy bumpy conditions best of the 4 types listed above.
Figure out what you really want, then decide on the rest.