Eu acho que você está perdendo o ponto. A ideia de Destruição Mútua Assegurada é precisamente por que o Capitólio não usou armas nucleares na primeira guerra.
During the Dark Days, the rebels in 13 wrested control from the government forces, trained their nuclear missiles on the Capitol, and then struck a bargain: They would play dead in exchange for being left alone. The Capitol had another nuclear arsenal out west, but it couldn't attack 13 without certain retaliation. It was forced to accept 13's deal. The Capitol demolished the visible remains of the district and cut off all access from the outside. Perhaps the Capitol's leaders thought that, without help, 13 would die off on its own. It almost did a few times, but it always managed to pull through due to strict sharing of resources, strenuous discipline, and constant vigilance against any further attacks from the Capitol.
-- Mockingjay, chapter 2 (emphasis mine)
Destruição Mútua Assegurada ("se você nos atacar, nós dois morremos") é o conceito exato sendo referido aqui. O Capitólio não matou Treze por causa de MAD - é por isso que Treze foi autorizado a sobreviver. Em resumo, a resposta para sua pergunta é:
How could the Capitol use nuclear weapons against District 13 provided that “Mutual Assured Destruction” doctrine was in place?
Eles não o fizeram, e precisamente por esse motivo.