Though I will say, the MP not saving when using Tents and Cottages, that’s a high priority bug that doesn’t directly affect gameplay, and that’s probably what I’d be interested in providing an optional patch to fix.
Tents work as they should though. As for Cottages, can we be sure that Cottages doesn't recover your MP before saving your game is actually a bug? Yes, it's stupid but this game have a lot of unusual design choices.
It may very well be a programming mistake, but the fact is that both the Item Menu choice/description and the Japanese manual support this quirk being by design.
Cottage manual description:ふだんは手の平に乗るほど小さくなる魔法の小屋。辺境の地で休むにはぴったりのアイテムです。ＨＰ、ＭＰは大幅に回復します。（ただしＭＰの回復はセーブの後になります。）