As probably the only one in these forums who's actually been in charge of a military installation, I've got some insight into the problem from both sides.
There are three major challenges facing a base commander before they can approve use of DoD property. And when I say DoD property, I'm NOT talking about property on a military affiliated museum or something similar located near the base but not on it. I'm talking about property owned fee-simple by DoD. Many of the clubs that think they're flying on DoD property are likely not, or if they are, they are just one letter to an IG from being thrown off (see below).
Taking the easiest first, it's actually the flying ops part of it. For a base like mine, the location had to be compatible with 250,000 military TACAIR flights annually, crop dusting operations on ag outlawed fields, other low altitude traffic traversing the airspace (fire, police, etc. helis), security concerns (weapons magazines, other sensitive facilities), and access. With very few exceptions, EVERYONE entering a DoD facility has to be either DoD affiliated (meaning they have a USG ID of some sort), or be sponsored by someone who has one. Most bases did not give what I'll call "categorical" sponsorships. I was expected (by my leadership) to tie every individual non-dod person to a specific DoD sponsor. The reason is simple: the DoD member is accountable for who they choose to sponsor and, to some extent, what they do while on my facility. In other words, in the event of misconduct, I'm going after BOTH of them. I'm also obligated to ensure that anyone for whom I grant access is of no threat to my base population (family in homes etc.). No felonies, no child related crimes, no dishonorable discharges, etc. So IF you make it past those wickets, the AIROPS piece isn't too difficult, however I do have some work to do with my AIROPS officer to figure out best way to deconflict. That usually involves a combination of geographical, altitude, and operating day and time limits.
The second challenge is policy, and that can rear its head at several levels. The overall DoD policy comes down to this: "Whatever you do for one entity, you have to do for all similar entities." So if I approve a private dues collecting organization to use my property, then a strict interpretation (usually the HQ's or even DoD level) would require me to grant permission to all other private dues collecting organizations that request it. Now I can try to limit that a bit by playing with definitions, but literally all it takes is a letter to the Inspector General that I'm playing word games to improperly excludes some groups. So when one considers approval, as the decision maker you have to consider the other groups that could apply that you really DO NOT want to approve. Here's an example I faced. A very good, forthright organization came to me with the best of intentions requesting to build a monument on base property. When I later sat down with my JAG, she said "If you say yes to them, you. have to say yes to anyone else. So are you willing to have a memorial to controversial figures? Or what about memorial for enemies of the US? Or KKK? etc. etc." Point being, you can't discriminate and get away with it. We said no. It's also entirely possible the higher HQ just says "no." Why? it's usually tied to the above, but at a regional level. If one base is doing something, it makes it hard for other bases to say no. Oh, and once HQ or IG finds out you're trying to skirt the rules, the club gets thrown off on short notice. I'm aware of at least one club that lost their access this way.
That said, the one way you can discriminate is to limit membership to only those with DoD affiliation. That means Active Duty, Retired, Dependents, Government Employees, Government Contractors WITH a DoD ID Card OR honorably discharged with DD214. I could say no all day long to entities that were not exclusively DoD. However, that's where AMA gets a little testy ... as they want clubs open to all members. Hence the inherent conflict between AMA's objectives and requirements of DoD.
Third is financial risk. While some in these pages tout the AMA insurance, and yes that has value, the issue facing DoD is different. When something happens, the one that gets sued is those with the deepest pockets. And DoD has deep pockets. So while AMA insurance is there, once someone gets hurt on DoD property the lawyers go after DoD.
My recommendation is to either create a club that's exclusively DoD affiliated (as described above) and then ask the base, or look for land elsewhere.