Why is the Pip-Boy so inconsistent when invisible?
Solution 1:
From my experience, the Pip-Boy only becomes entirely invisible if you just crouched or stopped moving right before using your Pip-Boy. It is likely a bug, where you're seeing the Pip-Boy's transitional state between being visible and invisible.
The only fix is to wait a little bit after crouching or stopping, until your character turns invisible, before you use your Pip-Boy. Naturally, standing up or unequipping the Chameleon equipment would be even better, but not necessarily what you want to do.