I agre to the second passport if you plan to do this much. I have done this numerous times and never had a problem with a paper stamp either. The one disadvantage is that when you present a passport with lots of Arab visas and entries/exits you may also have the mother of all security inspections; polite but oh-so-very-thorough. Still, that is not much of a hassle. I have never heard of the Israelis not agreeing to a separate paper when asked.