It’s...complicated.

First off, it would do very little for their relationship with Iran. Iran is Shi’a, and the Shi’a don’t recognize the authority of the Sunni caliphs anyway. The - I don’t know what I would call it, sub-denomination? - of Islam that is the state ideology in the Kingdom commonly referred to as ‘Wahhabism’ (they don’t use this name themselves), which is technically a derivative of the Hanbali madhdhab (one of the four major Sunni legal systems), considers Shi’ism heresy. So, the mufti of KSA declaring himself caliph would do very little to change the situation.

Now, the other thing is that the Saudis took over the peninsula in the late 1910s and early 1920s; one of the things they did was boot the long time rulers of the Hijaz (the western province that runs along the Red Sea; Mecca and Medina are here), the Hashemites, out of the kingdom. The ruler—the al-Saud’s main rival—was Hussein bin Ali, who had been lobbying very hard to become not only king of the Arabs (his sons Abdullah and Faisal would become the kings of Jordan and Iraq, respectively), but also to replace the Ottoman sultan as caliph. However, once he lost not only Syria but his own kingdom, his claim weakened (he didn’t have much support from outside) and he died in exile in Cyprus.

The question of who, then, would replace the Ottoman sultan as caliph was very political and discussed after the position was abolished in 1924 (to clarify, the Turkish parliament ended the monarchy, and then stripped the former monarch and his heirs of the title of caliph, leaving the position vacant—it’s usually described as “the abolition of the caliphate,” but that’s an oversimplification.) The Muslim community worldwide never came to a consensus as to who should replace him and the position remains vacant. Generally, however, it is understood that a caliph should be appointed by consent, which is why the leaders of ISIL and Boko Haram declaring themselves caliph didn’t have much of an effect on people not already inclined to take them seriously—that’s not how it works.

However, there may be another explanation in Wahhabi doctrine itself. They call themselves Muwahhidun (‘unitarians’ is a rough translation, another might be ‘followers of the one’) in which there is God and Qur’an, and nothing else. When they conquered Medina, they caused an uproar by demolishing the tombs of the Prophet’s family on the grounds that people were treating them like shrines and nothing should be worshipped but God. They were even preparing to demolish the Prophet’s tomb itself—on the same basis, people were coming and visiting it, and Muhammad is not an object of worship in Islam—but there was enough protest that they backed off from that.

Hence, the caliphate may well be a ceremonial title that bestows reverence they don’t want. Also, the king would not be the caliph. The al-Saud and the descendants of Ibn al-Wahhab have agreements about which titles and which ministries will be governed by whom, and introducing the title—which would, in theory, outrank that of King—would likely upset the power balance between them.

Someone who knows more about the inner workings of Saudi religious thought may have a firmer answer, but that’s my educated best guess.