The '98 movie really fits perfectly with GMK, and I think that was intentional on the part of GMK's makers. In the '98 movie, some Japanese characters, upon seeing the reptilian daikaiju in the film, think that it's Gojira/Godzilla -- which pretty clearly implies that an earlier creature called Gojira did previously exist in that film's continuity. In GMK's timeline, the real Godzilla hasn't been seen since 1954 (except for an unconfirmed, possibly mistaken sighting in New York a few years earlier), and the government suppressed a lot of the specifics about the incident to avoid making the Self-Defense Force look bad, so the Japanese public has forgotten a lot about Godzilla, and because of that ignorance, there are characters who, upon seeing Baragon, mistake it for Godzilla.
The only problem with that theory is that the animated Godzilla: The Series
, a direct sequel to the '98 film which was much better than the film, doesn't fit in the GMK timeline. Because the events of the show would've had to take place between the '98 movie and GMK, but the SDF commander at the start of GMK said that there had been virtually no kaiju sightings since '54 other than the New York incident, whereas in G:TS, there were kaiju cropping up weekly all over the world.