Personally, I've always assumed that Godzilla II died at some point before DAM, that Minilla grew up to be Godzilla III in DAM, and that the Minilla in DAM was a new character entirely. Or, alternatively, it's just a movie completely on its own without any connection to anything else.
It may frustrate some people, but it's much easier to understand Showa Godzilla as an "era" rather than a single "series" of films. It allows you to separate out the films, only a handful of which share any continuity with each other. Continuity and ongoing storylines didn't really become a major concern in this franchise until the 1980s.