You might just want to skip the car complication.

I taxied from Eilat Sheraton to border, walked across,
easily found taxi on far side direct to Petra for $60 I think, can't remember exactly.

Stayed 2 nights at Crown Plaza Petra, very nice, right on site, hotel desk arranged a continueing taxi to Allanby crossing, I think $80 one way, amazing sight as you drop down to the dead sea. Lots of ravines filled with tent-camps of Iraqi refugees. Express bus to Jerusalem for $15 or so from the border

This eliminates any possible hassle with a car, getting into an accident where the foreigner is always at fault, etc. We were almost the only car on a lovely 4-lane divided highway (it was Friday)!

And the taxi guy was very good fun, very familiar with the international situation etc.