There are specific rules in this case.
In the case of a trailer, you are allowed ONE (1) tent also. (not that the rules are followed or enforced). So you're fine.
The official Disney website, in its description of the Premium campsites, lists the dimensions ("on a concrete pad") of UP TO 18'x60'. (I call phrases like "up to", "possibly", "may be" by a clear term: WEASEL WORDS).
You are also going during the busiest time of the year (between Christmas and New Years) when the campground is normally completely booked. Therefore, you will be placed in a site that "most likely" (another pair of weasel words) has a guest departing the morning that you are arriving. So you don't have your pick of all Premium sites. You can only be assigned (naturally) whatever is available on check-in day.
Info in "The Book" (which does exist) has been on the web and apps in various states of having info updated, operating systems updated, etc. But since you won't know your site number until it is READY for you to set up, site specific data is useless IMO.
The best thing you can do is a 2-step process:
1. Call the WDW Reservation line and ask that a note be added to your reservation stating that you will also have a tent of "X-by-Y" dimensions. "The Book" has information for each Premium site if a tent has space to set up adjacent to the concrete (or if bushes, trees, utilities block a tent) on the dirt/pine needles.
2. Arrive as early as you can at the Fort, go into the lobby and check-in in person (or use the drive thru booths (don't use Online early checkin) and talk to the CM in person to reiterate your need for a Premium site that can also support a tent. The CM may be able to "sprinkle some pixie dust" to make it work. Example: you and me both arrive on the same day with a Premium reservation. The system randomly assigns me a Premium site that HAS space for a tent and a site for you that DOES NOT. As long as I/me haven't checked in yet and you arrive first, the CM can swap you to my site and me to yours and I/me, arriving later, will never know the swap occurred. But if I/me or others have checked in before you, the option to swap with us is naturally gone since we now occupy the site.
As a final word, I am an avid tent camper in all shapes and sizes. At the Fort I normally camp in my trailer but I am embarrassed to admit how MANY different tents I have. Tents where I sleep on the ground (that normally have stakes in the ground), canvas wall tents with a self-supporting standing frame used with cots, or some mix. Free standing tents (no stakes required) if that's what your friends have can stand on the concrete (tuck a ground cloth under the edges of the tent) give you leeway to use the concrete pad so any Premium site should work; but if they have a tent that needs to be staked out in dirt, then your choices become limited. You can only be assigned a site that is available that arrival day - and of all the sites available on arrival day, only some might be able to handled a staked down tent easily.
CM's do review the notes on campsite reservations before arrival day so give them enough info to help them do what you ask and they are super about trying to make it happen. I often add a note requesting a certain loop and I've usually gotten it and the few times I didn't, I drove around my requested loop on the way to my assigned loop and, sure enough, every site was already taken in my requested loop. So I believe the CM's do try.
Bama Ed