The most effective way of avoiding the crisis would have been if the Shah had died sooner (in summer 1979?) or else had ben in a less serious state of health at the time and had not had plausible need of treatment in the US but could have been dealt with by American (or French?) doctors flown in to treat him in a less politically inflammatory location - eg in a relatively stable Arab state that was not at risk of large-scale demonstrations by outraged citizens stoked up b the new Iranian regime's ideologists, eg Morocco. Or a European state that did not have large investment in the Middle East that was at risk of a boycott or asset seizures by Iran (or fury by its own, powerful anti-US leftist political factions) if it was seen by militants as aiding a 'deposed tyrant and US stooge', eg Spain.
France was unlikely to help the Shah for those reasons, and had its assets and its long-time reputation as a semi-independent actor in the Middle East to protect; Germany was too cautious and had a semi-Socialist (though moderate) govt with a need to keep its supporters onside; any Arabian peninsula (Sunni) monarchy with good hospitals and a suspicion of the new Iranian regime was at risk from its own Shi'a population in some cases (eg Bahrein) or military/ naval/ air raid threats across the Gulf from Iran whose navy could block its oil exports through the Straits of Hormuz. Oman was the exception here in naval terms as its ports faced the open Indian Ocean, but was still at risk of attacks on its Northern tip across the Gulf - and all the Gulf states, including Oman, had had dubious relations with Iran under the Shah in the 1960s and 1970s over his claims on Straits of Hormuz islands so they might not have wanted to risk Khomeini's wrath to help an ex-foe. Alternatively, if his decline in health did not occur so soon the crisis of where to treat him might have been postponed to 1980 - by which time the US was in election year and admitting him would stir up an unnecessary crisis that could backfire.
The US government and its intelligence services had been naive and poorly informed/ overoptimistic about the situation in Iran re: the extent of popular anger and anti-Americanism in the run-up to 1979 (with Carter notably regarding Iran as a stable and lauded ally on his visit there a few years earlier) so it might have misjudged the reaction to letting the Shah in in 1980 too and been faced with a major embarrassment and a hostage style crisis . But this would be a much smaller one if it had had an extra 6 to 9 months of withdrawing its staff, as would be accelerated by anti-American demonstrations and violent propaganda in Iran in their election year for the new Presidency, 1980 - when radical Islamic candidates could be expected to burnish their street cred with the public, the revolutionary committees and militias, and Khomeini by attacking the US and its local 'spies'. (If Bazargan or one of his top secularist allies, eg veteran liberal anti-monarchist 1979-80 Foreign Minister Yazdi, was running for President or PM in 1980 their radical rivals were likely to try to defeat them by playing up that they were 'soft on the enemies of Islam' by keeping the US embassy open and not throwing out all 'satanic' US influence). A turbulent struggle for the Presidency and for Parliamentary seats in Iran would likely ramp up anti-US outrage there, with the surviving semi-secular Bazarghan govt not daring to intervene as this would be seen as showing that they were 'American spies' too, and this might even lead to armed militants attacking the US embassy on some other excuse - and the Iranian govt lacked the reliable military and police allies in Tehran to protect the embassy, as the local commitees and militias had both well-organised groupings based on those created in the city during the revolution and lots of weapons stolen from govt armouries then.
Once the Pres election was over, if the relatively secular and left-wing but still nationalist Bani-Sadr (in exile in France in the 1970s and not hostile to Europe or Western culture) was elected as in real life he did his best to secure control of the army and acted as commander in chief from the outbreak of the Iran-Iraq war that summer, but his control of the new Govt and its PM (led by more anti-US Islamic radicals close to the emerging Islamic Republican Party and their organiser Ayatollah Beheshti) was limited and a power struggle between them followed in 1980-1. In that situation, Bani Sadr could not seem to be seen as protecting the American 'nest of spies' or he would be in trouble too - so he might have given the Americans a 'friendly warning' to pull out all their diplomats, perhaps to a less volatile place on the Gulf coast with an airport for quick exit if needed (Shiraz?) if a full evacuation was too drastic, and avert attacks on their embassy. We could still get the embassy stormed and/or some diplomats seized by gunmen from the Tehran militias, stoked up by Bani Sadr's or Bazarghan's enemies to embarrass them, but not a full scale crisis - and less risk of a US helicopter dash to try to rescue them, which made matters a lot worse and showed up US 'warmongering' and 'incompetence' at the same time. (This in turn probably encouraged the Americans' radical enemies elsewhere to attack their assets as their response was so feeble, eg Hezbollah in Lebanon - the Lebanese radicals were usually a lot more wary of attacking USSR assets as the Russians would be far more brutal and effective in their response.)
Carrying out a US military operation in Iran during the height of their own election would be too risky , unless Carter was desperate for a symbolic victory and listened to gung-ho advisers - though equally any perceived US humiliation, eg failing to retrieve a few remaining diplomats in Tehran in summer 1980 who had been kidnapped by gunmen and paraded on TV as 'spies', would be used by Reagan as evidence of Democratic complacency and incompetence and would add to Republican votes. Foreign policy is not usually crucial in a US election and George W Bush got away with his failures in Iraq in 2004 - but perception of a weak and blundering President facing a skilled, focussed populist strongman (and an ex-actor 'tough guy' sheriff at that) would hurt Carter.
The Iranian Pres election would pretty certainly have seen the failure of any surviving 'in office' candidate from the Bazargan govt as not patriotic/ Islamic enough with the voting urban and rural masses, and so a 'revolutionary hero close to Our Leader' winning (be it Bani Sadr as in OTL or another candidate) whatever had happened in Nov 1979 over the embassy crisis, and the new Iranian constitution would have seen the end of the interim Bazargan govt in any case pus the appointment of a more religious one allied to the radicals by Khomeini. The whole concept of the 'velayat al-faqih', ie direction of the political process by the Leader as the supreme theological authority, was at odds with the ideas of most of the former (nationalist and broadly secular) National Front party of the 1950s, the former party of Mossadegh and the home of his allies and heirs like Yazdi, anyway and at best only a few of them could have expected any, limited, influence in any govt appointed by Khomeini who had entirely another model of rule in mind - and now had it enshrined in the new constitution with a popular mandate. The more liberal , mostly semi-secular and semi-socialist elements in the new regime , led by Bani Sadr and the new Foreign Minister Qotzbadeh (one later deposed and exiled in 1981, one executed), were too weak to prevail against the religious elements and their rev. committees/ gunmen/ Rev Guard/ judiciary allies in 1981 anyway, and avoiding the Hostage Crisis would not change that - nor were they pro-American, or able to afford being seen as 'soft on' America.
At best, a lesser or avoided crisis yet rising tension and Iranian official rhetoric in 1980 might have made the US more wary of backing Saddam Hussein's invasion that year and so enraging Iran further, though SH was likely to have invaded anyway to get the banks of the Shatt al Arab lost in 1975 (and Khuzestan with its Arab minority?) back and the CIA to have secretly encouraged him to try to bring Iran down. The new Iranian govt would also have been , as in OTL, close to and noisily supportive of the PLO and Hezbollah (the latter fellow-Shi'a) who had backed its exile members in the 1970s, and this would impel a US/ Israel - Iran confrontation in the near future.
