One thing that is clear is that Harrison would to win the electoral college outright in 1892 or the overwhelming Democratic majority in Congress would strangle the Lodge Bill in its cradle. It is viable to get Harrison to a narrow majority by flipping the Deep South states with the worst vote suppression records and a couple of close and historically pro-civil rights Northern states:
One of the most surprising aspects would be the prompt end to segregation and Jim Crow laws in the Deep South states while these practices would presumably continue for many decades in the peripheral South where the 20-35% black vote wouldn't hold sway. If he won in 1892, Harrison would get 2 more SCOTUS appointments by 1896, so segregation would presumably still be upheld, but perhaps by 5/3 instead of 7/1. And the case wouldn't come from Louisiana.
Then comes 1896, which with Harrison as president, the Republicans are sure to lose. But do they lose to the Democrats or to Populists who have moved to the social right? Harrison probably moves further left on silver in his second term to peel off some Western Populist support, so the Democrat is probably pro-union but also pro-gold standard. Supposing a Democrat wins in an election with federally enforced black enfranchisement, he probably does this by dominating the Midwest and peeling off NYC metro support over Silver:
From here, Democrats probably hold the presidency almost continuously until the Depression. Given 6 years of truly free elections to consolidate black Republican power, I don't ever see Democratic machines being reestablished in LA/MS/SC/AL/GA, but the window for any further progress would close as Democratic-appointed judges soften their enforcement of the Lodge Bill. There would probably be substantial black migration into the Deep South states where they would have political power and full civil rights. Racist whites would presumably migrate out of these states after realizing that they could not muscle their way back into power. Republicans have to wait until the Depression to get full control of Washington. With more black political power to advocate against it, segregation ends in the 1930's either by constitutional amendment or by Republican SCOTUS appointments. 1932 would look something like this in that world: