OP, it's pretty bad form to solicit everyone's advice and hide the ball on MAJOR constraints. And as a regular FTer, you should know better than to waste everyone's time like that.
In any event, a day trip to the Dead Sea after a day trip to Petra is an incredible waste of time and a lot of backtracking. Plus, the Dead Sea isn't the best use of a full day when you have very few precious days in these two countries; there's not much to do other than put some mud on, go in the water, float and take a pic.
If you insist on spending the 3 nights in Amman for Hyatt status, I'd second craz's advice, stay elsewhere on night 2 (e.g., overnight at Wadi Rum), and hire a driver to take you to Petra, Wadi Rum, the Dead Sea, the Madaba map and Karak castle. And then I'd cut out the North from Israel and focus on Jerusalem.
Last edited by LAX_Esq; Dec 26, 2019 at 1:00 pm