"They" is a very good neutral pronoun to use for genderless pets as a whole, although some people do use one (one, oneself, etc) as a preferred pronoun!! I think it ultimately boils down to your pet's character -- there's quite a lot of different nonbinary pronouns you could use!!

One doesnt seem right. I have recently typed up a legal document for work and they had she/he/it written in it alot. I found that "It" was not a very PC way of speaking about someone that doesn't want to or cannot identify as either gender. Upon looking on google myself it seems like "they" seems to be the accepted term to use

Small Kodama walked and walked until dawn and when they couldn't walk any more, they reclined against a dead old tree.

It sounds right to me

In the case of Kodama, you could always use "it" since I assume that gender doesn't exist for their spirit type. Otherwise, using they/their would be fine since it's gender-neutral and more respectful of different gender identities or a lack thereof.

Using "one" is more common in Europe, but I was under the impression that it typically replaced the word "you" during times when the speaker is voicing an opinion or addressing the general public. For example: "You should always give proper credit." becomes "One should always give proper credit."

I've seen used to describe someone who is non-binary before, but it sounds a little awkward here.

Hope that helped!

Thank you for the feedback!! I'll stick to they ❤️