I only have a couple of my dogs marked as competitors, but I recently bought some gidgets that have been contested in the past and so they have levels on them. They are not marked as competitors, I have checked several times. They do not show up on the list to sign up for competitions, but they show up on the menu to challenging another pet. I'm not sure if this is a glitch, or if that is the way that is it supposed to be, I just thought I should mention it in case it is something that can be fixed.
Non-competetors can still battle, provided they have the levels for it. No glitch, that's on purpose.