If he thinks that anyone who wants a gun should have to undergo a psych eval, that's a whole different kettle of fish. Who gets to do the evaluations? Who gets to decide what a disqualifying factor might be? Who bears the cost? At what point does any of that begin to infringe upon a constitutionally-guaranteed right? How long before someone pushes the idea that "if you want a gun, it means that you're predisposed to violence, and therefore disqualified from owning one?"
Do you have to sacrifice whatever right to remain silent you may have for whatever right to keep and bear arms you may have?
Will you have to submit whatever papers from your doctor you may have trading some portion of your 4th amendment rights for your 2nd ones?
Will a denial based on psychological grounds be appeal-able? Reversible? Will it be in front of a judge to protect some form of Due Process? How much further would such a ruling go? Will anyone who fails this psych eval be deemed Adjudicated Mentally Defective theoretically losing their voting and freedom of movement rights? (And if not, why the hell not? If they're too nuts to have a gun, why do we want them voting and running around loose?)