OP- if your daughter had been assaulted and intimidated by someone other than your son what would you suggest she do? I’m reasonably sure that you wouldn’t be trying to smooth it over after a “stern chat” with her assailant. You realise she could report this as an assault to the police? I never quite understand why being intimated, assaulted/ abused by a family member is treated less seriously than the same actions perpetrated by a stranger by some.
Even if provoked- which wouldn’t excuse his actions, and it’s not clear that your DD did behave unreasonably to cause that provocation in the first place- he can’t go around physically assaulting people who make him angry or frustrated.
then consider your DD- I think it fair that she feels safe in her own home at all times (not only if her brother is out or one of her parents in the house too). If she says she is unhappy or upset or frightened of being alone in her home, then you have to take that seriously abd not try to minimise her feelings.
if, a future boyfriend, for example, behaved the same way when she tried to get him to hurry up in the bathroom as she was needing to get ready to go out, would you think “having a stern talk” or his mum giving him an ear bashing and a chat to try and “work out why”would be sufficient to rectify the situation? Or do you think you would be encouraging her to end the relationship, because if he can do it once then he can do it again? You have to treat this as you would if someone else had assaulted your DD and apply the same standards.
If your son gets away with these actions with very few (if any) consequences, it’s more likely he will do it again. Or escalate. By all means try and help him understand why this happened if it is out of character, but even if there is something going on for him which provides some mitigating circumstances, it doesn’t excuse what he did and it almost certainly doesn’t wipe the slate clean with your daughter. But he definitely can’t just say “all sorted” and expect that to be it. He needs to engage with you so you can help him fix any problems. You