I wouldn't bring her in then put her out again - change of temperature kan kill quicker than cold if they're used to being outside in the first place.
If she has plenty bedding and insulation, I'm sure it's not much difference her being in the shed than in a hutch. Is there no way you can block a decent sized area of the shed off for her rather than her being shut in the crate? I wouldn't leave her loose or unsupervised in the garden as you've no idea how bunny proof it is or what predators are in the area. Foxes are very bold with no fear of humans and she would be a sitting duck even if you were at home.
Personally if her new accommodation isn't ready yet, I'd take her old hutch just for now, give it a good scrub inside with bleach to help kill the mould then get some cheap lino tiles or an offcut and just stick it to all the inside surfaces to stop the damp penetrating. If there's anywhere she can get at edges, a bit of cheap wood battening over the edge will stop her ripping it up. It will cost between a tenner and twenty quid which is nothing in the face of keeping her safe.