While it's true that contracting and surviving smallpox confers lifelong immunity, and, given the somewhat more urban nature of the North there may have been more opportunity for exposure and consequent immunity to the disease, that was certainly not true for the average Southern soldier. Given the rural isolation of most Southerners, exposure to various diseases was minimal. For this reason, Southern ranks were periodically decimated by disease, measles in particular, especially in the Spring of 1862. Additionally, when rural Southerners were confined in Northern prisons, smallpox epidemics were, sad to say, almost routine. The U.S. Military Prisons in Illinois -- Alton, Camp Douglas, Rock Island Barracks -- were notorius smallpox centers. Hundreds, perhaps thousands, of Southern POWs died of smallpox while confined there. The Union Army, on the other hand, had a fairly extensive smallpox vaccination program. The concept of vaccinating against smallpox originated in the 1790s, although lacking an understanding of the germ theory of disease, I'm not sure that most people knew why or how the disease occurred, or why vaccinations worked.
I've studied the Civil War extensively for 25 years, and have never seen any references to the use of "bioterrorism" by either side, apart from the occasional poisoning of wells in the Trans-Mississippi Department. These latter activities seem mostly to have been the work of guerrillas (bushwhackers, jayhawkers) from Missouri and Kansas.
My sense is that if there was any bioterrorism, it was an insignificant factor in the war.