They shall fight and lose, same as OTL everywhere in the Americas, same as with any successor state of the European colonial empires, due to massive technological inferiority and limited numbers.
Here it is advocated a different outcome must happen, just because the ATL opponent is a slaver state, everything else being the same, even if militaries of OTL states with similar counter-insurgency-focused organization and similar social issues (peonage instead of chattel slavery - not that different) always won.
A different outcome "must" happen because the ATL opponent is a generally f--ked up mess with enough issues to overload the ship of state to the point of sinking before it starts trying to send its pitifully undersized and overstretched regular army out to chase Apache on their home ground.
The idea that the CSA failing to do what the USA only accomplished with difficulty is an Apachewank is kind of hysterical.