Honestly just make it random drop 1/whatever when using eggs on the shrine in the woodcutting guild. Once you unlock it, maybe you can transmog the color or you could use the other two eggs on it like with sarachnis to unlock the colors. Not sure how the community would feel about it but I think that would make it an interesting way to obtain it.
I believe there was discussions around the time the shrine was added to do roughly that! Adding pets has historically been quite challenging due to players concerns, perhaps that line of thinking is less of an issue these days :)