First, I hold with RichardInSF re your extreme limitation. This said - if you do want to stay at the Park Hyatt - I submit that your only reasonable options for seeing the highlights are the hop-on-hop-off or l'Opentour Bus. The latter's point of departure is a 10-minute walk from the Park Hyatt. l'Opentour is a double-decker with an open upper deck and earbuds with an English narrative. It has four tour loops or a ticket that combines them. You might drop your stuff with the bellman and walk on down to its office. Save time by looking online.

After that, you have somewhat of the lay of the city and can go back by foot or easy Metro to whatever turned you on.

I'd add a "moreover" to something else that Richard related: The Musée d'Orsay - one of my favorite museums in the world - is subject to pop strikes that you don't anticipate, don't know the duration - could last an hour or the day - and you only learn of them by an 8 1/2 x 11 sheet on the door or online. (They happen so often that there's a special web notification service.) So the suggestion is that the d'Orsay is for next time. And the queue always takes a long time at Ste-Chapelle unless the weather is awful. Next time. Conversely, you might come up on a line all around the place in front of Notre Dame - but it really only takes 15 minutes to get in. Sacré-Cœur should be one of your planned bus hop-off places -- basilica + view but an uphill walk from the Métro.

Eating is easy anywhere - and there are many reasonable restaurants within walking distance of the hotel.

OK, a lot of words for a very short stay. All toward efficiency.