@Phaaze 1. It's not a public bath, it's a private bath for the students. It's risky but also not too risky.
2. You don't know the layout of the castle. That could be a very public entrance where people walk by, but I highly doubt because it was a Royal bath. I doubt the average person in the castle will just walk by and be able to see the King bathing. It also could be a specific hallway that leads to the bath. Meaning only those 30ish people who would have a reason to be there would possibly come. (Students, maids, butlers) Of those 30 only half are a factor as they split gender bathing. So risky but also not too risky.
3. Time 24 hours a day and that bath is used for maybe 3 hours. Yes time wise this was the riskiest outside of use time. Right before in the filling time when Females of the 30 might come to check the progress.
And I don't know if you missed it, but they got caught. Ha manga logic, am I right?