Is there a downside to increasing ISO and decreasing shutter speed
With an A7S, MkI or MkII; the camera is ridiculously incredible in terms of noise hint even at extreme ISOs; that's compared to just a few years ago when everyone was shooting on the 5D MkII. 

My recommendation BECAUSE it's the A7s and because the visually acceptable level of noise to the viewer is really not noticed until you go into the extreme ISOs; would be to try to maintain a consistent ISO regardless of whether you are indoors or outdoors, or at least a fairly consistent range. 

Shooting at 6400 ISO will have virtually no noise, and if you shoot all your material at a set ISO you will have a consistent grain/noise hint to all your footage. Just keep in mind you gain one stop of light each time you double your ISO. 

Shutter speed can be adjusted, along with your aperture; to set exposure; along with ND Filters if you have them. 

You should really shoot some tests at various shutter speeds to find the look you are happy with. The standard for film/video is a 180 degree shutter; which provides a slight amount of motion blur when shooting at 24-30fps (48-60 Shutter). When you have a higher shutter; your footage will be more "juddered", that is, each frame will have a higher sharpness value with each step up in shutter speed. This can sometimes be a little distracting. 

A great example of this is the D-Day Invasion Scene from Saving Private Ryan; shot at high shutter speeds; there is virtually zero motion blur frame to frame; and it adds to the scene's frenetic; racing feel.