Sometimes when taking a late flight (ie redeye), a hotel will offer the room for an addtional 6 hours past the 4 pm check-out time for a fee....something less than a ful-night rate. Maybe a $60 charge in the US or maybe 50% of the room rate.

Maybe the answer is asking the Park Towers for the ability to check in at 2 am with only paying that 'fee'- ie not a full night rate...

Worth a try??