The refrigerator we rented 4 yrs ago at ASMo did have a very small freezer compartment. It was probably only big enough to accomodate one ice tray.
I'm not sure if they're using the same models, but this refrigerator was a small dorm-type fridge. It had a small "freezer" area that we used to chill gel-packs / blue ice for the next day. The gel-packs never froze solid because they were only in the "freezer" compartment overnight.
If your inserts are not too big they should fit. I would definitely have a spare set so one can be chilling while the other is in use.