I wouldn't recommend loft insulation as even the type that's wrapped in silver foil does shed, and the particles can be dangerous for bunnies just as they are to us. Personally I think I'd use a lot of bubblewrap sellotaped on to keep it from moving, then it's easily removable come spring. Then duvets, and then some waterproof layers on top.
You can get custom hutch covers made which you could get made a little big to allow for insulation. Lots of people get 'shutters' made to fit perfectly into the mesh parts of the hutch/run. They're often just a single piece of wood with some holes drilled in for insulation, and kept in place by some bolts or turny latch thiniges. Depending on the size of wood that you use, you could sandwich insulation between two pieces - meaning that when you had them in place, each of these holes would be insulated as well as everywhere else. Obviously insulating these parts affects the ventilation so you'd wanna make sure there were a couple of small holes for airflow.