Why didn't I think of this solution before? We're staying at the Hotel Alyeska in Girdwood the night before our late evening departure. I just talked with them and they allow late check out depending on availability....sometimes free, sometimes a charge. But, if they are booked, they told me we could use the showers in either the fitness center or the spa at any point during the day of check out.

Thanks to all of you who provided suggestions!