The most likely outcome i can see is that the new president after he is innagurated tells the south to shut up and quit complaining because they war.
A semi-non ASB option would be several southern leaders refusing to recogonize the new president, so he marches the army down south removes those dissenting politicans from office, ends a few riots and all goes back to normal.
Not very likely.
As I mentioned earlier, nine of the eleven Confederate States have
already shaken off Radical rule and got back to something like "business as usual". Are they really going to start another war, and risk having Radical governments reimposed on all of them, for the sake of Mr Tilden, whose inauguration would make little difference beyond hurrying things along somewhat in the two remaining States where Radical regimes are still hanging on by their fingernails? It's just not worth the candle.
After all Radical rule has little future even in SC and LA. The Democrats already hold the HoR, and are cutting off funds for military occupation in the South. From 1878, the regime change in the South virtually guarantees them control of the Senate as well. Reconstruction is already dead, and the only remaining issue is the best way to bury the corpse. Why jeopardise all that?
An interesting option would be to have a new presidental election with different candidates as not to offend either side of the election dispute much.
That possibility was seriously discussed in the New York Times. It raised the possibility that if inauguration day approached with the current deadlock unresolved, President Grant could resign a day or two before his term expired. Since there was then no Vice President (Henry Wilson having died in 1874) the President of the Senate, Thomas W Ferry, would become Acting President for the remainder of Grant's term, and would, constitutionally, be entitled to continue thereafter "until a President shall be elected".
If the deadlock over the 1876 election remained unresolved, a new election could be held in November 1877, under the provisions of the 1792 Presidential Succession Act.