GH4 isn’t some magical piece of kit that automatically works in every situation. Personally I wouldn’t record any cinelike/log-ish footage if the camera isn’t recording to at least a 10 bit format.
That said. Are you possibly using some sort of LUT as a starting point? It might be clipping out information before you can get to it. Have you tried the footage in Resolve? It has better secondaries so doing selective correction might be easier on that. It also has it’s own color management where you can change color spaces and gammas without using a possibly destructive LUT.