I don’t think they care whether they are legitimate or not to be completely honest. Hamas won the election in '06 fair and square, then defended against an attempted coup…whether that leaves them as a legitimate power or not is beyond my scope of understanding. One could probably also argue that they lack of elections for the last 18 years makes them illegitimate. But regardless, Hamas is an easy excuse whether they were elected fairly today or 18 years ago. I honestly don’t think Israel would treat Hamas and Palestinians any differently if they won a fair election again next week.
I can think of one immediately that may fight the US for that title…