You CAN put a bedrail on the daybed. I've seen pictures of it set up like that. The bed rails Disney uses are an older style, that clamp onto the side of the mattress. They are not the kind that "require" a twin sized or larger mattress, like all the ones on the market today.
In addition, the Poly daybed has "arms" which is good, because with a bedrail on, it will be almost like a crib (i.e. all enclosed)
I'd just do that. As a second resort, you can try the Pack and Play. We took our son to D-Land and he was 20 months at the time and he's a BIG boy! He's super tall and heavy (he's 37 inches now at 23 months). He slept in the pack and play just fine. He still uses one every day at home to nap. He is fine in it. He moves around a lot, but he SLEEPS curled up, usually. He doesn't sprawl.
As a final resort, I've heard of people folding up a blanket and laying it on the floor between one of the queen beds and the wall, and putting pillows down to "block off" under the bed, if necessary. Their toddlers have seemed to love having a little "nook" to sleep in, and as a bonus, since they're down lower, the light from the room/t.v. won't disturb them.
You have plenty of options. Something will end up working.