A post-bac is technically non-degree courses you took after a bachelors, and not part of a second bachelors.
Its kind of an in between area after a bachelors, but before a higher degree (masters, PhD DDS).
In the OPs situation it doesnt really matter what you call it because it is just a name, and wont make a difference anywhere or to anyone (except for the fact that you do have a second degree).