Thank you for sending your project! I have confirmed your project. First of all, I would like to confirm one thing, there is no need to display two different colors of lips at the same time, right? You may want to place the images of each color of lip in the skin placeholders and use linked meshes. Linked meshes may be under skin placeholders in different skins, as long as they are under the same slot as the source mesh.
In your case, it seems that the Mouth and Mouth2 bones are set as skin bones and it has a role as skin placeholders, but this is not appropriate. I think that is the reason for your confusion because if bones are set as skin bones, the timeline for the bones will not be displayed while the skin is not active.
To learn more about skin placeholders and linked meshes, please see the following pages of the user guide:
Skins - Spine User Guide: Skin placeholders
Mesh attachments - Spine User Guide: Linked meshes
By the way, although I did not point this out at first because it is not directly related to this issue, using mesh deform keys is not recommended due to processing costs. If you are creating a video work, it doesn't matter much, but I guess your skeleton will be used with spine runtimes, so it would be better to make the deformation using weights and bones as much as possible. To learn more about this, please see the following page:
Keys - Spine User Guide: Deform keys
