I do think you handled this largely well, but your phrasing that his brothers may need to know - I'm wary this could push an unsure child to make a declaration publically that he's not ready for and may not really be him, and I'd suggest if said in that way to privately clarify that with him that he doesn't need to tell anyone else to be himself.
His unsureness may not be a phase, he hasn't actually come out - he has a few skirts and thinks he might be trans. Maybe that framework will fit him best - he'll need to have discuss with a non-judgemental therapist who can see whether he has gender dysphoria and then how to handle that (not all dysphoric people transition or identify as trans), but that can wait if there is no sign of distress.
Maybe trans is just the closest concept to how he's feeling and would feel better with a broader range of options - some others have been mentioned with male role models who've worn feminine clothes or the rise in FemBoys in certain online spaces (which I would double check - sadly there are also spaces where boys with an interest in feminine clothes are targetted by groomers).
Maybe it's just a bit of experimenting for now and will pass with a bit of time and understanding that it's perfectly normal to play around with social roles as a teenager to figure things out. He doesn't have to attach his sense of self to this, as popular as that is now.
There is nothing to tell his brothers at this point and, more important, he's just as much himself no matter what he tells other people or if he chooses to present one way with some people and another way with others. The wide social message that people are only themselves if everyone else knows their sexuality or how they view their identity is frankly horseshit. Some individuals do like it all out there, but just as fine are those who are private and careful with information. I don't think children should be given the message that they will be less themselves if others aren't told, it's an impulsive age and I think with teenagers in this day and age, cautioning on controlling personal information has benefits.
With how things are today, I also thinks its important if it comes up again to try to clarify what he means by trans and how feels it might apply. There is a wide range of definitions out there now, and it's good to be on the same page even when that page may change at times and generally having an attitude that it's possible, but nothing needs to be set in stone yet.