Both eyes are going to stay wide open when time slows down. In the weak hand, the pistol has to cant so the sights find the dominant eye more easily. The head will move a little to help the dominant eye get there, too. A little practice starting from off hand low ready to dominant line of sight will be all that's required for the arm and neck muscles to solve the problem. Try dominant and weak hand low ready to weak eye line of sight, too, keeping both eyes open. The dominant eye will fight this, and if there is enough time, from a secure position, to make a long range shot, it should be closed. Putting tape over the dominant side of the shooting glasses simulates eye out, without the discomfort. Actually, a woman had an eye gouged out by her BF and said she didn't feel that much pain.
Shooting weak side cover is best with weak hand and eye. The head and gun both are canted, and separation from cover allows case ejection. Avoid the natural temptation to push the head out for use of dominant eye. The same applies to the peek. The cover will occlude the dominant eye, much like the tape on the shooting glasses. Find a way to peek high and low, and alternate as you pie a corner. Or peek high and shoot from low in case they are set for your next appearance. It's never a good idea to show from the same place or make a sound, and don't use a laser or light until the ball drops.
Some of this can be done at first w/o shooting. Find out what works best for you and choreograph it. Smooth is fast. Eliminate anything unnecessary or awkward, except from cover. If your comfortable, your exposing too much.
Any eye or both will do for point shooting. Try taping one side and then the other of your shooting glasses. Try this while point shooting with the off hand, too. When both eyes are kaput, we're down to taking sound shots and recon by you know what.