Still having a very hard time finding a name for the mare. Here are a few more I found.
Kachina - from Hopi or Pueblan culture - a term for the spirit beings that inhabit natural objects (trees, sun, rocks...that sort of thing)
Faline - name of Bambi's friend - little doe
Kitsune - means fox in Japanese
Pixie - fairy or elf
Sprite - elf or forest spirit from midieval folklore.
I'm sure I'm going too deep with this, but nothing stands out yet.