Roosevelt was a pragmatist more so than a statist - he was willing to try things. That said, it would be much harder for him to get elected in 1932 as a Republican, given that the GOP was in charge during the worst economic crisis in history. The easiest way to get a GOP FDR into the White House is to have an ineffective Democrat take over in 1932 and do such a poor job that the people would rather have the GOP back, run FDR, and have him win.
If that happens, a sick FDR would probably be pushed to step aside in 1944 and have someone like Thomas Dewey finish the war, this preserving the two-term tradition and invalidating the need for the 22nd Amendment, which may come later.
In this scenario, the GOP becomes the party of pragmatism and likely has more of a foothold in civil rights. Eisenhower was undecided on party affiliation for a long time, but in 1952 he may decide he has a better shot as a Democrat. He steps down in 1960, the Dems splinter as do the Republicans, and it's possible that we see the GOP as the party of big business and civil rights while the Dems end up as a populist labor party. Also, if Nixon stays GOP, his VP service is averted, he doesn't get elected in 1968, and Watergate never happens, averting a lot of the mistrust in government. Reagan never capitalizes on this and can't get elected in his very conservative platform. Taxes drop in the 1980s but not as drastically, the religious right never gains a strong foothold in the GOP and becomes a fringe faction of the Democrats, and the 22nd Amendment never passes but is never needed.
Of course, if FDR goes to the GOP, we likely never hear from him except as governor of New York or maybe as a senator.