Thanks for the additional information.
Call up a Connect Object, after the Hair Was cached.
Drag the Connect object under the Parent Null called “paintbrush baked animation”. Then the Hair object under the Connect.
Select the Connect object, and use
Object Manager> Object> Bake Alembic…
This works here.
Yes, if you don’t place this under the Brush, as I mentioned, then the numbers will be different of course.
It should not flip vertically, to begin with, in no case. To me, this looks like Priority glitch. (Hence the Connect idea above.)
Screen capture with the export version
All the best