Suggestion

Make them not use the default texture and meshes for mods that thange those.. or maybe by making the tiny ones a standalone entity with the same behavior and spawn rates as before...

This is better fixed on their end. Just shot the a message, will keep you updated here.

If I were to fix it all resource packs would need a custom texture for my mod which is not what I want, it should work out-of-the-box.

Looks like compatibility is not really possible. And btw, baby skeletons already are separate entities, this problem comes from them using the vanilla texture of the adult variant. But as described in my last message changing that would break compatibility with all other resource packs.

Fixed in Tiny Skeletons as of v5.0.0 for 1.19.3. The mod is now compatible with Better Skeletons.