Lots to discuss here. First, I strongly recommend going to a professional that serves the shooting community. It is absolutely possible to get what you have discussed. I use and highly recommend to clients Morgan Optical, and have had great service from Tom Morgan:
https://www.morganoptical.net/These new glasses will not change eye dominance, which is another issue. Eye dominance can be accommodated, but not changed.
A deep look at eye dominance may help. Eye dominance is simply the eye the brain tells to do most of the optical work. It sometimes changes with age, but cannot be changed via medical or other adjustments. While the subject seems on the surface to be cut and dried, it is far from it. It is not so simple as "right", "left" or "co-dominant", as there are degrees of dominance. The test for this I prefer as a long time shotgun shooting instructor is standing about 10-12 feet away from the client and asking the client to keep both eyes open and point their finger at my nose. The relationship of their finger to their face will tell a lot, and tell the degree of dominance (or co-dominance). Some people's fingers are just slightly favoring one eye, while others are much further toward the dominant eye. If the finger stays in the center of the face, the person is visually ambidextrous or co-dominant. This is important information to know before a person begins to shoot, as sometimes the tests are wrong and true dominance becomes apparent in the shooting (a good instructor can "read" the shot). There are several other tests of dominance, but this one gives more data in terms of degrees of dominance. And BTW, I and others have found that about 55-60 percent of women are cross dominant, while about 30 percent of men are. Who knows why?
What to do about the situation is another matter and there are several paths here. First off, the OP is rifle shooting and I would strongly suggest staying on the current/off shoulder from the dominant eye, as folks much past youth have a hard time with the switch. Especially if the point of this is hunting, I would suggest squinting or fully closing the off eye during the shot; same for handguns. I would absolutely avoid obscuring partially or fully blocking the off eye in a hunting scenario.
In shotgun shooting, presuming that the shooting bears out the cross dominant condition, my first move is to have the client close the off eye (if able; some folks can't). This will normally work, and it can be closed late in the process, such as when the gun comes to cheek in the move and mount in the field. In target shooting, I prefer to obscure a small portion of the off eye by placing a small piece of translucent (Scotch) tape over the spot on the glass to obscure the muzzle of the gun. This normally works, and there are some terrific shooters who do this (multi-Olympic gold winner Kim Rhode is one of them).
This will not always work, however. I have had three clients whose brain could actually see through the visual barrier such that the eye over the gun could not take charge. In these cases, the only path was to switch shoulders (and that worked, BTW, despite the uncomfortable period of building new muscle memory). All three were dedicated to the change, and felt as if someone let them out of jail as they began to crush targets confidently.
Again, this is a very complex subject, and much is trial and error in getting to the right solution. Sorry for the length, and I hope some of this is useful.
