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There's nothing wrong with going into Jordan on an Israeli passport. I know (second/third-hand) people who have done so.
Israel and Jordan currently have full diplomatic ties.
Chances are, if you try and hide things, they'll get more suspicious. Several years ago, we traveled US-England-Turkey on American passports, and from Turkey to Israel (also on the American passport), intending to use the Israeli passport only to get in/out of Israel. In Turkey, however, they did ask to see our Israeli passports. So you see, better to be totally upfront and use the same passport in both places (unless you have a good reason not to, such as when entering/leaving the passport's country).