Like Piggies&buns I would also let them choose unless it is below 10 degrees. Before we extended their hutch into the L-shape this summer, it was a 4 x 2 like yours (if I remember correctly). Ours worked out as even smaller on the inside and I would rather not have had to shut them in the small space overnight (necessary, of course), so for them to have had to be inside for 24 hours would (I think) have been too much. Having said all that, though, our two are like yours and do much prefer to stay in their hutch! They'll run out for a quick nose-around in the morning and then back in the hutch for most of the day, only really coming out for longer periods if we are there. They must still feel too exposed in this run. The bulk of the hay is also in their hutch (just a couple of heaps in the run). Having now experimented with three different run set-ups in the shed, it has been the smallest one they liked the most. I think if you do keep it to a compact run, and one with a lid as you say, perhaps it would be easier to keep it more snuggly? You could throw a blanket on top of that when very cold? Very many people on here have also stapled radiator insulating foil to the walls of their shed (
@Piggies&buns?) of which there are photos in this thread.
As for bath towels and mats, I don't really know. My fleece liners last year didn't quite cover all of the run, so in the remaining section I put down an old cot protector (a rectangle of terry-towelling type thing), then a towel, then a cut-off of thin fleece. The towel and cot protector did stay wet, but the fleece ontop was ok. Others have more experience of layering like that, though. However, if the piggies are not going to go out much, it is only going to have tiny patches of wee here and there!