Since I'm an Air Force airfield manager and this is part of my job, I'll give you the correct answer instead of pure speculation. Keep in mind this is for the AF, but all DoD installations will be similar. All the information for applying is in AFI 10-1001, but as mentioned above you'll need to submit DD Forms 2400, 2401, and 2402. It also says you should submit it at least 30 days prior and I'd say most places wouldn't approve it on a shorter time frame unless there were extenuating circumstances. This is due to the Major Command being the approval authority unless it's been designated down to the installation commander. Also, remember that military installations are official business only, so it's unlikely that they'll approve a CALP unless it's for an air show/static display, or another very good reason (such as you're a current service member and use your plane to commute to work). According to the reg, staying in government lodging could technically qualify, but that doesn't mean they have to approve it.
Your first step will need to be calling the airfield ops number listed in the chart supplement and ask if they currently accept civil aircraft landing requests (the installation commander may have a policy with blanket no's). If your request is approved then the permit is good at that installation until your insurance expires, or 2 years, whichever comes first (unless it's a single use permit) and you'd just need to request a PPR for each visit. You'll also be required to file a flight plan and give a copy of it to the airfield ops section before you depart the installation. Good luck and feel free to PM with any more specific questions.