You are not going to get anything predictable as far as sex links with that cross-breed rooster. It just does not work that way. About half his female offspring will be barred. Abour half will not. About half his male offspring will be barred. About half will not. That is if you cross him with a hen that is not barred. If you cross him with a hen that is barred, they will all be barred.
If you cross him with the Black Australorp, half the male chicks will be solid black and half will be barred. Half the female chicks will be solid black and half will be barred. With any of them, there could be some red or gold leakage, which means a few feathers could be red or gold instead of black.
If you cross him with the BSL hens, both parents of the BSL being RIR over BR, you will get a lot of solid black males and females. You will get a lot of black barred males and females. Any of these could have leakage. But you will also get a lot of chickens that range from white to many different shades of red and with different patterns. The white ones will have black tails and possibly black around the neck. The red could range from a Salmon color to a dark red. Half of these others will be barred and half will not.
I'm going to stay away from Buff Orpington predictions. Strange things go on when that buff is crossed with the black. Rareroo has some real pretty photos that demonstrate that. But they will not be sex links.
Whatever you cross him with, they are going to be mutts. They will not be sex links in any way. We all have our different goals, but personally I'd cross him with all your hens and expect to get a lot of different looking chicks. Since he is a cross, they are not going to be predictable anyway.