Alan:
Bad idea to trailer with the rotor on. Shock loading in the kinds of long trips you are thinking of doing can reach humongous values. Can you do it. Yes. Is it smart to do that. Not in my opinion. Shorter trips on good roads its more advisable. Taking the teeter bolt off, pulling the rotor up with soft ties using a top mounted arm and winch is not so hard. Then you can set the blades system intact (instead of taking 2 blades apart) on top of the trailer with proper box and cover on it. Your trailer needs to be about 24 foot in length. Folding mast will easily get you in standard trailers whose door height is 7 foot or slightly lower.
With the right setup you can take the rotor system off and put it on in less than 15 minutes. We can help you put the arm and winch on your trailer if you decided to go with us. We can mock it up in SolidWorks and get the swing arm for the winch laser cut and welded for a nominal fee.
In reality you have to stick with small airparks or airstrips. Unless you have permission from the land owner and he has no neighbors, you really can't just take off of any place you want to. That will get you in trouble with the law. In the US there are over 19700 airports, 5,700 of them are completely open to the general public. With a little planning there is no need to do the flying from illegal land spots.
To be honest if your purpose is to go poke holes in the ground in back country with big tires, I hate to bust any gyroplane pilot's bubble but we who have experience in other categories of aircraft as well easily know no in production gyroplane is as good and safe and easy as a bush STOL airplane for that purpose. None. Not a single model from any manufacturer. What gyroplanes will be good at is with some experience you will be able to fly in moderate conditions where it may be uncomfortable for winged aircraft to fly. So you have to define the parameters of your mission and choose the tool that fits your needs best. I know despite your thinking you are most likely to fly from airstrips if you are going to be in the US. Unless you are Burt Reynolds. Some Americans have always been a little crazy.
I do not know of places renting gyroplanes right now but that doesn't mean there aren't any. There are some Primary Category certified gyroplanes that AutoGyro produced that could be used for renting but practically the real limitation is going to be insurance because its not available for that purpose. But may be I am wrong.