The Ext balance isomorphism is independent of resolution comparison data

For fixed resolutions, the two edge maps defining the balance zigzag are induced by the augmentations $P_{\bullet }\to M$ and $N\to I^{\bullet }$, so they involve no comparison lift. After replacing a projective or injective resolution, choose a comparison map over or under the resolved object. These maps give a morphism between the two Hom double complexes and commute with both edge augmentations.

Any two projective comparison maps are chain-homotopic, and any two injective comparison maps are cochain-homotopic, by the two comparison uniqueness theorems. Applying $\mathrm{Hom}$ turns either homotopy into a homotopy of the corresponding total-complex maps. Hence the induced maps on all three cohomologies in the edge-to-total zigzag are independent of the chosen lifts, and the balance isomorphism is independent of those choices.