If your rifle has a magnified scope, you're going to close one eye anyway. So, in that case, cross-dominance isn't an issue at all.
The #1 benefit (IMO) of keeping both eyes open is for focusing on 2 things at once -- the front sight with one eye and the target with the other. There's also a tactical advantage of keeping both eyes open: to have the widest degree of peripheral vision so you're able to see other threats if present.
You can close the non-aiming eye initially to get the sights lined up with the aiming eye, then open the other one. This forces the non-dominant eye to already be focused on the front sight before the dominant eye becomes involved. Since the rifle is on the right side of your face, your eyes really don't have a choice of what to focus on anyway, so training them to do what you want isn't as hard as with pistols, where you can align the pistol for either eye to be in the sightline.
There's so much info on the Internet addressing this issue. Shouldn't be hard to find recommendations that work for you.