Freezer burn is caused by air getting to the frozen item. So, if your frozen rats are loose in your freezer and are touching the sides of the freezer itself, then yes, they will most likely become freezer burnt. Not because they were touching the sides, but because they were improperly packaged.
When you put your frozen rodents in the freezer, make sure they are in a proper freezer bag and as much air as possible has been removed from the bag. (DON"T suck the air out!!)
As for live feeding in the UK, I was under the impression that if the animal is starving and refusing all attempts to feed it F/T prey, you can get permission to feed it live. But I don't live in the UK so may have misunderstood.