Y'know, I think I used to eat Round Table kinda regularly when I lived in Vegas. The thing is, I haven't lived in Vegas since 2000. (I have spent a lot of time wandering the American West over the last several years, though, which means I've walked by countless Round Table units. But I never really had the money to buy Round Table's pizza throughout my travels, because I was functionally homeless.)
When I lived in Vegas, I never really thought about what makes Round Table's pizza any different from any other pizza (although I'm sure every bite mark would get a picture if I went there today). From what I think I remember, it seems like their pizza had a slightly soft, crispy crust and was cut into narrow slices. Very crispy, almost crunchy, on the bottom, but much softer above. Does that sound remotely accurate?
Even though your dough is a tad softer than my Tommy's dough (which is the pizza I showed in my pictures a few posts back), I'm still totally with fazzari here. And I know he has a ton of experience with this kind of stuff.
Interestingly, this is actually something I've had to deal with a good bit lately because I've hosted four pizza parties this summer, in which I've made as many as four different styles of pizza (NY style, Tommy's style, Malnati's style, and my first Giordano's style experiments). Which means I've had to put a lot of thought into how I manage all these different styles of dough over a period of basically a whole day, as if I was operating a pizzeria. In fact, my pictures a few posts back are from one of these parties.
The Tommy's style (cracker) skins are the only style of skins that required considerable planning and advance prep.
Like you, I had to roll my cracker style skins well ahead of time. Unlike you, though, it's because it takes me at least half an hour to roll and trim each 14-15" Tommy's style laminated skin (8 laminates). I would do this at 10:00 or 11:00 in the morning, then put the 2 or 3 skins in the fridge, removing the first one at 1:00 or 2:00. What I can tell you pretty confidently from this experience is that you shouldn't have any kind of problem if you only intend to leave the skins in the cooler for a matter of hours, rather than days.
Do what fazzari suggested, even though you won't necessarily need to. And if you ever plan to keep skins in the cooler overnight, definitely do what fazzari suggested. I prefer to use my skins immediately after rolling, or at least within a few hours, but I think fazzari has said he prefers to use them more like the next day, which I've also done. Even if the skins dry up a little, they still usually come out fine.