inspector should have known that at this time of the year is not good idea to be without inner cover if nothing else - because of lack of top entrance, which is very important when bees work the sugar feed, (getting rid of excess moisture)
This teen-test is also called " Homasote " but is lately hard to get, without asbestos/silica added. Needs to be pure paper/wood mixture, kind of fluffy, (it wicks out moisture which forms where warm and cold meet)
You can look for this stuff in lumber yard, Home depot?
I could not get it this year here in Canada. So I was looking for old-fashioned ceiling tiles. They used to be made from this stuff. But they are also now full of glass wool and silica which is carcinogenic for you and not good for bees.(made in China)
I have, by chance, found in the pre-cut section of the lumber yard, this 2'X4' pieces of CORK-board ! The kind used to pin on notes in schools and offices. It's about half inch thick and you cut it on table saw to the size of the hive. It's a bit more money but once you have it it will last forever...
About the move? It will have to be your call. I wouldn't move nothing when bees are in a cluster, is very stressful for them. When temp is around 60 is OK, but I personally do not open nothing below telescopic cover, this time of the year! Cause, bees have now propolized all the cracks and crevices for winter! If you break it up they will be distressed cause it's too late for them to get propolis and fix it up. They may get chilled if it gets windy!?
I thought so, that it was a typo about sugar mixture...
Don't worry too much, you're doing OK. People here will turn you in the right direction. Just like they say: "Two heads are better than one!"