First, the Black Panther Party was not a street gang at its inception. They did not peddle drugs, guns, or sex at their start. The Panthers were a self-determination, self-reliance black nationalist group at its core. They didn’t really turn into gangs and all that whatnot until the 70s. The Ku Klux Klan was not simply a gang, but a terror organization. They were two distinct entities, with two dichotomous missions. One was about uplifting the black community, and the other was about scaring the black community in the South into continued subservience.
They likely wouldn’t come to blows unless the Panthers decided to pack up and move to the Deep South, or if the Klan decided to show up in the inner cities.
Now what you could have happen is offshoots of the BPP showing up in Southern states, and going after Klan-affiliated sheriffs and politicians. You could similarly have fire bombings of BPP offices, and possibly shootouts, but nothing to the effect of gangland-style shootouts.