The second Godzilla movie, Godzilla Raids Again, was kind of like that. It featured the first kaiju-on-kaiju fight, Godzilla vs. Anguirus, and neither one was the "good guy" or the "bad guy"; they were just giant animals acting on instinct and barely even noticing the humans underfoot. And a number of the Heisei- and Millennium-era Godzilla movies had Godzilla as a bad guy/threat but had him take on an even worse threat and be the lesser of two evils, or at least the devil we know -- the execrable Godzilla vs. SpaceGodzilla, G vs. Destoroyah, Godzilla 2000, G vs. Megaguirus, Final Wars. I guess Gamera vs. Barugon, the least dreadful film in the original Gamera series, fits the pattern too, though Gamera's initial badness in that film is limited to the recap of the first film and the token destruction of a dam. You could also make a case for Gamera 3: Revenge of Iris, the conclusion of the pretty awesome '90s Gamera reboot. Gamera had been a hero in the first two films but kind of lost his way due to the events of the second, and there's a really potent, chilling sequence where Gamera fights an enemy monster in the Tokyo equivalent of Times Square at its busiest time and carelessly causes enormous loss of life.
But that doesn't really fit what it looks like we're heading for in the Legendary Universe, because both Godzilla and Kong are being established as hero kaiju. So that's different from a situation where both are villains, and it makes it harder to justify putting them into conflict. Legendary Godzilla has been defined as a creature whose evolutionary purpose is to be a check on more destructive monsters and preserve the ecological balance (much like '90s Gamera, in fact, though naturally evolved rather than genetically engineered by Atlanteans). So if Kong is supposed to be a "good" monster, then presumably Godzilla would recognize that he doesn't pose a threat to the natural balance. Which would make it harder to set up a "misunderstanding"-type excuse for a hero-vs.-hero brawl.