I tried that. Did SS 2x a week and 1x was doing BW stuff. Doesn't work. You need the rest on SS. (Admittedly, I'm 40, so 2x a week is enough!) After a few weeks on SS, when you're putting up the weight, your body needs its recovery time.
I don't see a problem mixing barbell training and BW stuff, but you can't do SS and BW at the same time and do either justice. Perhaps pick one modality, do it for 6 months, then switch.