Is there a way I can equip Doop with his cam? I have the common doop, but since he's hidden, I can't equip him directly while he's hidden. I also can't transfer the cam to him since you can't transfer equipment to hidden characters. Anything I can do???

You can achieve this by bringing Doop into the visible area. This can be done with a number of cards, including Overexposed, Out of the Darkness, and Shadow Step to name a few.
Once in the visible area, Doop can work his publicity magic.