How do I get these little feather scallops to stop curling?

The wetting before punching is very likely most of it. When you soak leather and then compress fibers with a punch, they dry back in whatever shape the pressure set them in, and a round punch pressing into a damp piece tends to dome the leather around each notch. That cupping is what's making your feather points flare. Try punching everthing dry, or if you want soft cut edges, dampen only lightly with a sponge after punching and then press the piece flat under a book or a board with weight on it overnight while it dries. Pressing while damp is what trains the leather to stay flat. Hammering against a sandbag won't hold because the leather is drying back to its set shape with nothing constraining it.

On my scalloped card holders I found stitching the liner on actually solved most of the flare. Once the two layers are cemented and stitched togehter the leather has no room to curl, the seam kind of locks everything flat. The tails on yours are probly thin enough to skive though, I take mine down to roughly half thickness on anything with long points and they behave after that. A slicker and some gum works the feather tips over too, gives them a bit of stiffness so they hold thier shape instead of flopping around.

Gotta push back on the whole 'press it flat under a board overnight' idea. I tried that exact thing on some scalloped key fobs and the points still popped right back up once the weight came off. Small notches hold way too much tension for passive pressing to do anything. What actually worked for me was skiving the back of each point thin, like paper thin at the tips, before stitching. Once the tips have barely any thickness left they've got no stiffness to curl with. Punching damp or dry matters way less than getting that thickness down in my expirience.

That third reply matches what happened on my owl pouch. I pressed everything flat and the wing tips still curled by morning, so I skived the backs of each point instead. I did it with a scalpel on a glass slab, just shaving from the base of the notch out to the tip so it feathers down gradually instead of stepping off thin all at once. Took maybe ten minutes a wing and the flare was gone for good. The long tail points are the same deal, just work slower near the tips since they tear easy when they get that thin.

Reading this made me realize why the pointed corners on my bookmarks always flare out. I never connected it to punching them damp, I just assumed my cutting was sloppy. I ended up gluing the corners down against a backing piece which sort of hides the problem but the tips still lift if you look close. One thing I'm wondering after the skiving suggestions, doesn't taking the tips down paper thin make them fragile? My bookmarks get bent back constantly when a book gets opened, so I'd worry about them snapping off. Maybe less of an issue on a coin purse that doesn't flex in the same spot over and over.

Update: followed the scalpel-on-glass advice and shaved the back of each point thin, tapering toward the tip. Wings lie flat now even after a couple days. Stitching the liner on helped too, the seam realy does hold everything down. Tails still needed the most thinning but they've relaxed. One tiny tip still wants to lift and honestly it just reads as character at this point.