I think that one of the easiest ways to have Rhodesia survive would be to have it gain dominion status earlier. In 1939 there were talks to federate both Rhodesias into a dominion, but the war intervened.
Something else that could have made Rhodesia last longer is to have a larger European population. This could have been remedied by having the the Rhodesian government be more open to immigrants. Up until the 1960s it was incredibly hard to settle in Rhodesia even if one was British.
A good example is in postwar Britain there were thousands of Brits who applied to migrate to Southern Rhodesia alone. In 1948 over 80,000 British from the United Kingdom applied and 4,300 were admitted.
Instead of trying to attract more British immigrants, the Rhodesian government made it harder by placing more restrictions on perspective immigrants. Starting in 1948 for a Brit to settle in Rhodesia he would have to possess at least £1,500 or have a regular income of not less than £500 per year (the per capita income in the United Kingdom was less than half that, about £240). The logic being that Rhodesian should be a gentleman's colony. Still 4,300 immigrants arrived from the UK that year. In May of 1952 further restrictions came into place with quotas.
A maximum of 10,800 White Immigrants were to be admitted per year. The annual quotas being as follows:
United Kingdom and Ireland 4,560
British Subjects from the Union of South Africa 4,560
British Subjects from other African colonies 600
British Subjects from Elsewhere in the Empire 360
Aliens 720
Nevertheless, the European population increased as follows
1901 11,000
1921 23,600
1931 49,900
1941 68,600
1946 82,000
1951 135,600
1961 221,500