I've wanted to speedrun ff1 but I only have mobile. Should there be a new category, or will it fit in another "glitchless category" such as psp

Usually each console has their own category due to various subtle discrepancies in game play. Are you looking at running it on an emulator on your mobile device or is there a specific version of the game for mobile that you're looking at? In most cases new categories are added as community interest grows so if there's a demand for it then it'll usually get added.

I also have the mobile version bought from the App Store. I too want to do a bestiary run but not sure where it would fall under when it comes to the platform.