Hyatt is encouraging trips, not nights. And they think a 2nd free night is mostly just a cost with the 1st free night being sufficiently motivating to take the trip.
This is an easy limit to work around. Nights 4-6 in wife’s name. The clerk won’t have a problem changing name and WoH number upon final checkout - a nice smile is probably quite sufficient. You do know it’s not the revenue manager himself doing the checkouts?