I agree that it would start with requests to local militias and then the governor. The situation sounds similar to
Cincinnati's 1841 race riot, though that riot began with less specific provocation. In that case, militiamen from the surrounding area were called in to get control.
Another similarity would be the 1863 Detroit riot:
http://docsouth.unc.edu/neh/detroit/summary.html It too was handled in the state, though Detroit's first full-time police force was formed soon afterwards.
That's ignoring, of course, the obvious parallel to a riot that got out of control, the New York City draft riot, where federal soldiers did need called in. In that case, it seems Lincoln had no problem calling in soldiers when needed, but if I'm understanding the different premise here, in the hypothetical example the rioters would be primarily blacks staging a massive attack of the surrounding white neighborhood.
One problem I can think of is lack of numbers. Were there any northern cities with a large enough black population that they could attack like that and not be put down by local white ruffians, Irish, and other natural enemies? Add in police, sheriff, militia or home guard, and I think it would be hard for blacks in any northern city to get enough momentum to get such a riot going. But maybe I'm wrong, or am missing a better similar example of such a thing.
As far as the specific question about Lincoln... Depending on his relationship locally, he might try to get a respected black leader or white abolitionist to address the mob and call for peace, but if that wasn't practical or if it failed, I don't think Lincoln would have a problem sending in federal troops if they were needed, as in the New York City riot. The alternative would be letting the death and destruction continue simply because black people were doing it. I don't think many people short of your John Brown types would have the stomach for that.