Would successor Veep-Congress harmony be enough for Grant to feel no need/desire to run in 1868? Since it seems a GOP nomination would be his for the taking. Holt or Dix would be opposed by Seymour playing the race card like in OTL? Neither Holt nor Dix would have made a buffoon of themselves like Johnson, but neither would have Grant's star power - maybe Seymour would have a better chance?
It is possible, certainly. That said, Seymour would still likely defeated by Dix (I personally prefer him over Holt) though potentially by a narrower margin, seeing as many would still see the Democrats as the party of the Confederacy. Come 1872 though, I could see the Democrats winning the election due to the Credit Mobilier Scandal (I think ITTL, William Franklin would take the nomination, as due to a stronger TTL Democratic Party, he would not feel the need to step aside for Greely).
That said, it is likely that Radical Reconstruction applied continuously since 1865 would effectively snuff what would become the Democrat Redeemers in their cradle, particularly in states with a black majority population. This, combined with the possibility that Confederate emigration (e.g. the Confederados) would likely become more widespread ITTL would mean that the position of newly enfranchised freedmen in these states would be solidified, preventing the rollback of black civil rights that we saw IOTL. Something rather interesting I discovered about James Garfield on Wikipedia:
During Reconstruction,
freedmen had gained citizenship and suffrage, which enabled them to participate in government, but Garfield believed their rights were being eroded by Southern white resistance and illiteracy, and he was concerned that blacks would become America's permanent "
peasantry".
[185] He proposed a "universal" education system funded by the federal government. In February 1866, Garfield and Ohio School Commissioner Emerson Edward White drafted a bill for the National Department of Education. They believed that through the use of statistics they could push the US Congress to establish a federal agency for school reform.
[186] But Congress and the northern white public had lost interest in African-American rights, and Congress did not pass federal funding for universal education during Garfield's term.
In the (admittedly somewhat unlikely) event that Garfield's education bill passes, I could imagine that within a couple of generations, an African-American population that is better educated would be better able to stand their ground against any lingering efforts to restrict their rights. Additionally, given that it is unlikely that Dix or Holt would overturn
Special Field Orders, No. 15, and it is possible that this may be extended in other parts of the South; if this happened, it would mean that TTL's black population would be far better off, politically and socially.
A scenario I've played around with is having Ben Butler as Lincolns second VP. He was a political player who would have no problem playing ball with anyone.
My idea is that an amendment giving congress the power to appoint a new vice-president if the office is vacant is passed. Partly because Butler is hated by the south so much, the fear there would be attempts on his life inspires this. Grant is elevated to the vice presidency soon after. When 1868 rolls round, Butler runs with Grant under the National Union banner.
One small issue with this is that, despite being an ex-Democrat, I doubt Butler would appeal that much to Northern War Democrats, partly due to his radicalism and partly due to the scandals surrounding his occupation of New Orleans. I doubt Lincoln would seriously considering him, given that if he really did not care about appealing to War Democrats, he could have just kept Hamlin (or maybe elevated Colfax?) in order to avoid all the controversy surrounding Butler. Additionally, come 1868, the Democrats would likely have a field day attacking Butler's antisemitism - this issue plagued Grant's own 1868 campaign and would be made much worse by having Butler and Grant on the same ticket, especially considering that unlike with Grant, there can be no questioning the fact that Butler was a very vocal antisemite.
Overall, I think this scenario is pretty unlikely (though admittedly fun to think about - for example, given his later membership of the Greenback Party, it is possible that Butler fights to keep greenbacks following the Civil War rather than reverting to the Gold Standard, which would have far reaching political and economic ramifications).