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If a laymen were appointed Cardinal, he would indeed be tonsured, but not even the minor orders were required.  But clerics are not per se members of the "hierarchy".  1917 Code of Canon Law required that Cardinals must be at least priests.  But even being a cleric is not required by divine law, since the Candinalate is not of divine institution.

"Hierarchy" is not a univocal term.  It can mean jurisdictional hierarchy or ecclesiastical hierarchy depending on context.  In the absence of context it should be assumed that the more general term, ecclesiastical hierarchy is the intended meaning.  The ecclesiastical hierarchy consists of all Catholic clerics, regardless of orders.  The jurisdictional hierarchy is more specific and consists of all those clerics possessing an ecclesiastical office with ordinary jurisdiction attached to it.  I don't know how relevant to the discussion this information is but I brought it up because I think a misunderstanding of the definition of hierarchy is a contributing cause of the current crisis.  SVs are getting hit with the accusation that the theory would mean the destruction of the hierarchy.  The jurisdictional hierarchy is indeed greatly diminished if not completely destroyed.  But the ecclesiastical hierarchy remains.  And the ecclesiastical hierarchy can restore the jurisdictional hierarchy if they put their minds to it.

In either case, is it somehow incompatible with Divine Law to have a lay Cardinal?  I don't necessarily see that.

If by "lay Cardinal" you mean a man appointed to a Cardinalate who is not yet ordained, yes, it has happened.  It even happened for the papacy.  But the man cannot receive the office until he has received tonsure.  After which he can immediately exercise the jurisdiction attached to the office (even before he is ordained).  But to call that a "lay Cardinal" is an abuse of terms.  He is just a Cardinal and all Cardinals are clerics.  You can make a distinction between a Cardinal who has no orders and one who does have orders.  But neither one is a layman.