They will not recognize the other country (palestine) because of religious/political reasons, for many years now. Womens ride or not.
Also, yes, two passports is best, if you tear out a page it will become invalid.
You can get two passports. Specify why you need 2 (or another) and it will be noted and considered for application notation.
My husband is going to be away for work and I can't go.
I think it would be neat.