How to keep thin sponge bases moist?

Is there a reason you're not just baking a standard thickness and splitting it into layers? This is going to give you more control.

If you have troubles moving the layers without breaking them, have a piece of cardboard handy (traditionally a cardboard cake round would be used by any would work) and after cutting the cake horizontally, lift up the edge of the top piece to slide the cardboard under and then proceed to push it all the way under the top layer before carefully removing the layer. Proceed with the next layer.

If they cake is a bit dry but not crunchy, moisten the cut surface by brushing with simple syrup on its own if you don't want any added flavor, or add a bit of a complementary-flavored liqueur to the syrup before moistening.

The type of pan that you use to bake your sponge in can help determine how evenly it will bake. I like to use a very flat, thick bottomed non-stick sheet pan lined with parchment paper or silpat.

I would also look to see if your oven racks are tilting the pan in any direction (this can shift the sponge batter to one side and make it bake unevenly or be more prone to crispy edges), or if the pan has the tendency to warp as it is heated. You might even hear the pan "pop" and that is obviously not a good thing. Does your oven heat evenly and hold a steady temperature? I know mine doesn't so rotating my pan during baking to ensure the center cooks evenly is key.

Otherwise, how you spread the batter to the corner will matter as the less batter you use, the harder it is to keep it from deflating. Try to use 3 to 4 strokes of a large flat offset spatula (you can purchase these at any baking/cooking store) to get the batter into the corners, no more than that, and then tap the pan on the counter to get rid of any bubbles (if there are any). I think that you also might want to experiment with scaling the batter up ever so slightly (like 10-15%) so you have a bit more wiggle room with deflation of the cake in the oven. You will still have a very thin layer, but perhaps more batter will keep those edges from crisping up. I'd be tempted to just scale up the egg whites for more volume when folding those into the base, but every recipe is different, so you'd have to experiment.

It is normal to expect that you will have to trim some of the sponge after it comes out of the pan with a very thin layer, but if it is more than 1/4-1/2 inch then something else is probably going on.

A light syrup is what the pros use, so I wouldn't worry about it making anything too sweet.

If you have a "Cake Spinner", one trick to get even layers by cutting is to put your rounds on it, hold the knife steady, and turn the cake, instead of the knife. Works pretty well for me, and I'm no surgeon either.

How about thinking outside the box for a solution. Try a using a microwave sponge cake recipe this one is from Ferran Adria. The microwave should cook it fast enough to eliminate the dry spots on the edges. It should work great with such a thin cake you're making.