On the subject of films never showing what Godzilla eats, isn't there a scene in Destoroyah where a bunch of dead whales are found? Also, let's not forget the scrapped footage from the original film of Godzilla eating a cow. I'm not saying that the general consensus isn't that he consumes nuclear fuel only, but there is precedent for him actually using those teeth for something other than fighting. If we're going to be sticklers about biology, you need to pay attention to the carnivore teeth.
I am of the opinion that Godzilla's biological classification falls outside of known taxonomy. He's not a reptile or an amphibian. There's no established rule that he is classified as either of those, thus making any argument about the gills being out of place on a reptile rather silly to me. Instead, I offer you a very realistic and believable approach for justifying the existence of gills: pharyngeal slits. All chordates exhibit this feature during development. In the case of Godzilla, they can apparently turn into the structures we see in the new film. Are they gills? Maybe. Are they pressure release? Maybe. Are they simply a scale structure? Maybe. We don't know what they are yet. When we find out, I expect it to be one of those things and I find any of them to be entirely believable. Sure, it's a feature that has never been on Godzilla before, and I can respect fans being irked over that, but to say it's a poor design choice because of taxonomy makes no sense to me.