It's very strange, I keep all mine in bat covers and one of them gained an entire ounce in weight. A few have gained no weight and some in between.
While humidity is the key factor, there are others. Even if a bat is oiled, if it's dry it will still want to absorb moisture from the air. A bat with a facing sheet will absorb (and lose) less moisture than one with a natural face. Likewise, big stickers act as a barrier.
The willow will have a natural moisture content that it will gravitate towards in your particular conditions, and some bats will start at a different moisture %, which is a key driver in how much weight they gain or lose.
I wouldn't have thought storage in a bat cover would make any difference unless the cover got wet, although it would slow a bat from drying out.
I mentioned in another thread that I did an experiment with a bat with a natural face and minimal stickers, storing it without a cover in an old (non working) fridge.
It started at 1145 g, I put a tray of water at the bottom of the fridge and left the door closed. 35 days later it had gained 60g (2.11 oz)
After the tray of water was removed, it dropped those 60g in 39 days. 4 months later after winter it was still only 1155g.