Also something I'm curious about. If my blues were lavender, would that be better for breeding away from black?
If you want the blue color, it would be worse. If you don't want the blue, it's the same issues as the B/B/S blue bird. Lavender is a recessive gene so once you breed a Lavender to something else you lose the lavender for a generation. If you then breed back to Lavender or breed Lavender mixes you can get some lavender birds. You will still pretty much have black or lavender birds.
The only dominant colors that affect black that I know of are dominant white or the blue/black/splash genes, unless you consider a black barred different. Sometimes you can get some interesting results if you cross buff with black. That often gives you a black bird with with some yellow feathers. Some of those can be striking.
If your goal is blue or green eggs from non-black hens, cross your EE rooster with your legbar/leghorn cross hen. You should get a mix of colors/patterns and about 3/4 of the pullets should lay blue or green eggs. What color is the legbar/leghorn cross? If it is white you are back to dominant white over black. I'd still use her but your results will not be as varied as if she is not white.
If you cross your EE rooster with any other not-black and not-blue hens you could get different color/patterns and about half the pullets should lay blue or green eggs.
If you cross your Black Ameraucana rooster with any not-black or not-blue hen you will pretty much get black chicks. But if you cross those offspring abut 1/4 will be non-black and most should lay blue or green eggs. Or cross the female offspring with your EE rooster. Again a rainbow of colors/patterns and maybe 3/4 blue or green eggs.
In all this I'd avoid the Rhode Island White just because I don't know her genetics. If she is dominant white you are back to that black base color and white chickens. If she is recessive white you could have anything hiding under that white. It might work out for you, it might not.
I agree that if you are not very careful after a few generations of inbreeding you can get birds that start to look alike. To slow that process down do not keep any black or white roosters, they tend to overpower other colors. Keep at least couple of roosters that don't look much alike. And keep as wide a variety in colors/patterns in the hens as you can.