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You will get several colors from that cross. Its getting late for me so my mind is shutting down a bit, but ill try to give you some rough estimate and I'm goin to assume the RSL hen you are using is red with a white tail but if not, let me know and ill adjust it.
A BSL rooster is only going to have one copy of barring so when bred to a non barred hen, half of his offspring will be barred and half will be non barred.
A RSL hen (if white tailed) will only have one copy of dominant white, so when bred to a non dom white rooster, half of her offspring will get dom white and half won't.
So you have to combine those to aspects together to figure the colors
Also, if breeding a barred roo, especially if he was bred from a red father, if breeding him back on red hens, the red is going to start dominating the extended black.
So with all that in account, the possible colors you could get will be
Barred (ranging from black barred to red barred)
Black (ranging to almost solid red, like the above color without the barring)
White ( ranging to almost white tailed red like the RSL hen, same as the color above with white replacing the black)
Red barred with all black replaced by white.
White with faint ghost barring. ( I have pics of this that ill post in a minute).
ETA. Here is a Ghost Barred cockerel that I bred a while back from a BSL roo on hatchery "RIWhite" hens. Its a result of one copy of barring and one copy of dominant white. Also, this cockerel should most likely be pure silver based since he has no gold leakage which would have been possible from his fathers side, however since you will be breeding to red hens, I would suspect yours to have some gold tinting and probably autosomal red showing in the shoulder region similar to a RSL rooster.
https://www.backyardchickens.com/forum/uploads/37373_w_bar_roo_2.jpg
Close up of the faint barring
https://www.backyardchickens.com/forum/uploads/37373_w_bar_roo.jpg
Thanks for the info... I set the eggs at noon today. I can't wait to see all the colors I get!
Good luck with them, id like to see what all you get too.