The problem with this scenario is that Ike never mobilized a mass movement the way Reagan did. He ran as an outsider, was really bad at politics (he had a relationship with Congress that could only be rivaled by Jimmy Carter, while he believed that kicking Nixon off the ticket in 1956 would be a better way to set him up for 1960), and by-and-large got elected because of his personal popularity as the hero of WW2. As far as his policies go, Ike was resented by no small part of the GOP; indeed, Goldwater, the 1964 nominee, even referred to his policies as a "dime store New Deal." He was also much more liberal than the GOP at large: the bulk of Republican leaders and voters had a philosophy similar to that of pre-5th Avenue Nixon -- i.e. centrist, but still conservative. Contrast this to Reagan, who ran (and governed, to a much lesser extent) as a conservative; he left a lasting impact, casting away an already-diminishing moderate faction within the GOP (see the 1976 primaries - Reagan earned 46% of the vote and nearly won the nomination from the hands of an incumbent President, a far better performance than Goldwater in 1968 -- even though Reagan's main challenger was much more conservative than Rocky!).
I should, however, note that Ike himself never viewed himself as a liberal or a Rockefellerite. He hated politics and ran because he was worried that the 'isolationist' Taft would win the nomination, and indeed tried to cut a deal to the very end in order to prevent himself from running. Much of the pro-business attitude that Reagan had could already be seen with Ike, whose Cabinet consisted of "nine millionaires and a plumber," with General Motors CEO Charles E. Wilson as Defense Secretary and conservative Texan banker Robert Anderson as Ike's dream successor, while Ike even did well in the South like Reagan eventually would. Ike was never a fan of deficit spending and tried to present himself as a pragmatic/moderate conservative, but was ultimately not an ideologue and recognized the realities of the time were such that anyone who tried to overturn the existing Keynesian consensus would find their career destroyed. Reagan was more ideological than Ike was, but even as President he was willing to understand that the realities of the 1980s were such that you can't just cut down the government to pre-New Deal levels. That was why much of his economic plan depended on a form of military Keynesianism, and why he never fully applied supply-side economics.