I know you want to find a purpose for him and whatnot. I feel like the easiest way to do that is to begin collaborating with someone so you can bounce off ideas rather than working alone.
I also know he is/was in the Kido Corps, so,
I was wondering if you would like Gekkou to be Kanaya Hizakura's initial mentor in the Kidou Corps after she was admitted from the Shino Academy? I'd imagine someone from the Shino Academy isn't going to be a full-fledged member from the get-go so she'd need a mentor to teach her advanced Kido theory and practice.
This way we can bounce ideas off each other regarding repertoire between the characters and can flesh out each other's histories and such as well.