He wouldn't have. Ike had a genepool Plains-German dislike and distrust of the Democratic Party; the only reason he was seen as a potential Democratic substitute candidate in 1948 was that this was not commonly known at the time. I don't agree with the idea that he wouldn't have any issue with Bush or with Trump at all, but I think he would chart a course much closer to someone like Romney or Liz Cheney than to the Lincoln Project if he were alive and politically relevant today.
Yea, I agree with this analysis, though it is hard to say just how some historical figure would react in a future environment. One thing I have been pressing back hard against lately is the underestimating of political divisions in periods of "political consensus". These always come across as down playing the divides that rose up to fill the vacuum or a generally dismissive stance towards a previous policy position because such is not well regarded by the mainstream today.
Just because a lot of Republicans were of moderate temperament and "accepting of the New Deal", doesn't mean that every Republican could have just have easily been a Democrat.
Eisenhower was not a Conservative Republican. He intensely disliked the GOP Western and Midwestern conservatives in the party. He hated Sen. William Knowland (R-CA) and was happy when he left the Senate to run for CA Governor in 1958 and lost. He generally did not think much of the Republican Party in general. His favorite politicians appear to be Southern Democratic conservatives, and he personally cultivated them. If you look at the brain trust of the Eisenhower Administration, you will see that most of the people who were "somebody" in the Administration were very singular partisans; the only Republican they cared about was Eisenhower.
Another myth was the idea that Eisenhower would have been more dovish on Vietnam. Despite his dislike for the Western conservatives, Eisenhower endorsed Goldwater. One reason was that Ike was hawkish on Vietnam. He was invited into meetings with LBJ on Vietnam in 1965 and in each of those meetings he was unreservedly hawkish and urged LBJ to go for victory.
Eisenhower, himself, was something of a racist. He was not a Lincoln Republican on Civil Rights. It was Eisenhower that first cracked the Solid South in 1952, carrying 5 states in the peripheral South, and it was Eisenhower that put the brakes on integration. He was sympathetic to the South's desires for more time to integrate, although he did send troops to Little Rock. Eisenhower was not a segregationist and did not support Jim Crow, but he was willing to advocate slowing the pace of integration in the South. I suppose that was part of his legislative strategy; Southern Democrats controlled many key committee chairs in both Houses and Eisenhower wanted his program to get through.