Hi Slicepong,
That’s some really good point you bring up there. So firstly it’s definitely good to try and keep fairly relaxed overall grip, but as you mentioned try to keep the index finder stable and not let it drop down. The thumb and index finger should apply most of the pressure of the grip to the bat as you correctly said but it can be hard to maintain and overall loose grip and relaxed wrist.
Fore the forehand I would say you can keep the same amount of pressure with the index finger and thumb, but on the backhand the pressure of the thumb becomes more important to add stability and control, especially hen blocking.
It seems to me like you can go for slightly tighter grip to make it more stable and stop some of the inconsistency and accuracy issues you said. Try to focus on being comfortable with your grip and not changing it too much, the finger and thumb are important for stability but so is having a grip that works for your shots, everyone is slightly different and one of the main things is that you feel you can play your strokes well without compromise. Hope that answers your query and gives you some more consistency!