This, if it occurs, will be a headache for whoever is President during this time.
I agree. I think that Iran would initially have descended into open conflict between the left-wing and the religious factions if the Shah was overthrown in 1963 that might have lasted into the late 1960s. I also feel that Ayatollah Khomenei would have joined the Arab Coalition in the 1967 Six Day War despite Iran's religious and ideological differences with the Arab states. Having Iran intervene in the war on the side of the Arabs definitely would have denied Israel a victory in the war and would have likely convinced the Iranian people to support their new government much like how the Iran-Iraq War did in RL.
It would have served to boost the Mullah government's popularity at home, yes. Long-run, this has massive butterflies in Isreali politics; Labor likely holds on power for longer.
It is also possible that Iran's presence in the Arab coalition during the Six-Day War would have resulted in Israel's destruction, as Iran has the largest and most well-equipt military in the Middle East during the 1960s in RL. Even though an earlier Iranian Revolution would have weakened the Iranian military to a point (I assume many of the military members connected to the Shah would have either been executed of forced into retirement by the government of Ayatollah Khomeini), Iran's military would have still been formidable.
I also feel that people such as Ali Shariati, Ebrahim Yazdi, Mostafa Chamran, and Mehdi Bazargan would have been major figures in the new Iranian government in this TL.