I am mainly logging this to track it, I may see if I can raise a PR at a later time unless someone gets to it first.
Unfortunately due to the structure where the name does not exist outside of the lazy-loaded on-click tooltip I can not see an easy way to get the Mount/Minion names in CSS Selectors alone for userAgentDesktop. For the time being it looks like the clearest path is to update the README for clarity.
meta.json Paths
profile/mount.json Before: https://%s.finalfantasyxiv.com/lodestone/character/%d/
profile/mount.json After: https://%s.finalfantasyxiv.com/lodestone/character/%d/mount/
profile/minion.json Before: https://%s.finalfantasyxiv.com/lodestone/character/%d/
profile/minion.json After: https://%s.finalfantasyxiv.com/lodestone/character/%d/minion/
HTML Structure / Behaviour
Minion
On Page-Load:

On-Click (of thumbnail):

Mount
On Page-Load:

On-Click (of thumbnail):

CSS Selectors
TBD