Very dangerous idea. One reason why the priest sex problem became so severe in the 70s was precisely because bishops agreed to turn problem priests over to the psychiatrists.
The psychiatrists didn't give a damn about Catholic principles, and basically they just told the problem priests to do what they felt like doing. Pederasts were quickly released back into the system to do their thing. Huge mistake.
Psychiatrists were also heavily involved in counseling several liberally inclined religious orders, and helped to destroy them with feel-good group therapy sessions.
Bottom line: don't trust a psychiatrist to do a bishop's job.
Where we really get into trouble is sometimes we can't even trust the Bishop to do the Bishop's job...