I find my RX100 has a delay on shooting sometimes; specifically when shooting anemone fish.
I don't believe it is shutter lag, which seems very short.
It seems to be that the autofocus hunts for focus.
What settings do you use for shooting still photos of potentially moving subjects. Specifically how do you set the camera to fire when you push the shutter, wether the camera thinks the shot is in focus or not?
Have you got a focus light on your set up? The RX100 works with contrast detection so it needs to believe there is enough light to go ahead
Alternatively you can try manual focus with peaking in that case the camera fires away immediately