To summarize, this is what I was doing wrong. On my presentation I tried to acquire the dot quickly, drive the dot to the target, grip the bejesus out of the pistol to minimize dot movement in recoil, help that further with post ignition push, then drive the dot to the next target and repeat the process, using strength to attempt to minimize dot movement.
This is how I changed. While I clamp hard with my support hand, I way backed off with strong hand grip. I look exactly at my aiming point on the target. I bring the dot to that spot on the target and fire the shot. I let recoil happen, then I use my peripheral vision to guide the dot back to my spot on the target. I shoot the streak of the dot returning into the scoring area. Then I snap my eyes to the next target spot, and when the dot reaches the spot, I fire the shot. Let recoil happen, and let peripheral vision bring the dot back to the aiming point. Repeat.
Instead of trying to minimize dot movement, I am embracing the streak. My grip is focused on making the dot return predictably and not minimizing dot movement. My split speed is faster because my trigger finger is less tense and I am shooting earlier. My accuracy is better because I am laser focused on my aiming point, and I can call my shots easier because I am focused on the target not the moving dot. Shooting on the move is also far easier with a complete target focus. After a practice session, I am so much less tired mentally and physically. It is fun.