Can I save my progress in the tower?

I wouldn't exactly call it a "secret", but there is a way to heal in between bosses during the tower.

I was having a lot of trouble without being able to heal, so I tried just going back to the ship.

If you go into the Odyssey, you'll heal back up. If you don't actually leave the dark side of the moon, the bosses won't respawn. They make it fairly time consuming to go back and forth, but you can use this method to heal between bosses.

As mentioned by ArrowCase in the comments, there is also a life-up heart hidden in one of the vegetable stems just outside the tower. You can go back and get it between fights. I have not verified this for myself.

There are no checkpoints on the Dark Side of the Moon, so you can’t save because the tower is meant to be a repeatable challenge. One of the bunnies on the moon even says you can redo it any time.

If you want healing anytime, you can use any of the 3 Peach amiibo or the Cereal amiibo to heal to 6 hearts. As much as you want & as often as you want.