The trouble is, Federal bureaucracy was pretty small back then. You need something which has more of an impact on everyday Americans - plus the South would have problems anyway accepting it.
Now, what you might see is in the 1930s, an already integrated buraucracy growing under the New Deal, so that millions of poor people obtaining relief see that the Feds are working together racially and think, "Huh, maybe it can be done."
It will take time, but the idea will be there faster, so that, once the end of World War Two comes, at least the NOrth is a good deal more ready.
Part of the question is going to be, does the Federal government also integrate the military? If they do that before WW2, even just a little bit (say the percentage it was integrated by the time of the Korean conflict), then you'd habe the races having fought side by side and being more comfortable with and trusting one another more. As Martin Luther King said, paraphrasiing, the reason people don't get along is they don't interact with each other anywhere.
(I don't know what they were thinking when they had Happy Days with Howard having a black Army buddy from WW2 - but at least they could have been at Stalag 13
like I wrote here.)
Even without that, you'd still have the New Deal, WW2, and so on as a place where the Federal government could be an example to others. I think without an integrated military also, you might get just a more peaceful '60s when it finally does come. With it, I think you could see it pushed up a decade, with the Civil Rights Act of 1964 becoming that of 1954, for instance. And, instead of people saying it took a Southerner (and a President's death), people would say it took a general who had seen it work in the military during the war to get it passed once he became President.