I do not truly have cross dominance.What happens with me,after 30 years of machining,and a whole lot of touching off cutters,I got sort of ambidextrous eyed.Figured that out from missing everything with a shotgun.When I put a shotgun in front of my right eye,the left eye took over.
A solution that works for me(maybe not you) is to let my thumb naturally stick up at the sky on the left side of the forend.It does the same thing your dot does.It gets in the way of my left eye,so my right eye maintains dominance.
Try it,see if it helps.
Having something on my lense drives me batty.I might rather learn to shoot left handed.
For a dot,I wonder if dry erase whiteboard marker would work.On glass,it would.Plastic,might be a danger of harming the lense.
Another option,go large.Like a WW2 blackout vehicle headlight or an Eskimo anti-snowblind hand made slit sunglasses.Put a horizontal bar of tape across the whole lense that you can peak over the top of.
I get eyestrain crosseyed crazy with a spot or a gob of Vaseline on my lense.