I've run into this with several clients, so I'd like to offer a couple of observations.
1. Closing the non-dominate eye has some downsides:
a. it limits target awareness
b. it increases facial muscle strain
c. it causes a lost of visual acuity of the open eye as the brain perceives there exist a lower light environment
2. It is highly unlikely that you'll willingly close either eye in a high stress situation, it is much more likely that both eyes will widen
I shoot with both eyes open, I just don't pay conscious attention to the image from the non-dominate eye. This is usually trained, much the way we train a lazy eye, by obscuring that eye. We usually do this with translucent tape.
...because the journey is the worthier part...The Shepherd's Tale