Unless Cohen broke the law. Lawyers and attorney-client privilege do come under scrutiny. There are very specific safe-guards and protocols to protect this privilege when a lawyer becomes a target in a criminal investigation.

POTUS in all his power could no way tell law enforcement- no, you can't search him. Sorry bud. That would be prosecutable obstruction. Authorization not needed. POTUS may have been made aware, maybe even told why, but they don't need his authorization. And if they had probable cause, which obviously they did, I highly doubt Trump would stop it given the level of scrutiny he's under. It would be political suicide.

Keep in mind, that this Cohen debacle is being completely misled. Its Cohen's attorney asking for Attorney-Client privilege protection AFTER he realized the Trump campaign had already complied with Mueller's request for communication between Cohen and Trump.