I have one of those (mines a 5 tier and no wheels). I keep it in my laundry room where I can close it off from the rest of the family (I too have small ones who like to try to pick the new plants in an effort to help me "weed"). I heat the room just barely below the temp of the rest of the house. I think if you keep it in your sunroom you will be fine. I haven't tried to grow seeds in mine yet but I have cuttings flourishing in it. I even got roses to take in mine. I'll be trying seeds in a couple of weeks.

Maybe you could clothespin a heavy-duty lawn/leaf bag or a contractor bag across the bottom of the plastic cover?

or stack several layers of cardboard under the "trolley"?

or roll up or wad up some sheets or towels & stuff them all around?

I used one in my sunroom in the past. It really keeps the humidity in and the temps are definitely higher inside but I never measured to see how much. I wouldn't use it outside yet - not enough protection. It's good to use outside for hardening off in late April/early May and then putting the cover back on at night. Keep the cover open outside during the day or your poor babies will fry.

I have one, they work great for starting seeds and hardening them up both...but watch the night temps, with the open bottom it can still get mighty cold in there..I thought about buying a heat mat this year for my fall seeds swap seeds.