Does anyone know the specific, required conditions to get the cut scene?
Crimsonrider summed it up pretty well I think.
A bit more anecdotal info:
IIRC, she won't initiate the dialogue the very instant you get her down to the required HP - it also has to be her turn.
On my current play-through, I had her down to 12 HP or so, expecting her to start negotiating as soon as her HP were low enough. But she needs to be up, or at the very least you need to end your active character's turn.
Not sure which of the two it is.
Since I play a Vengeance Paladin and didn't want to break my oath, I had Karlach with me during the fight (Barbarians have the best intimidation lines). When the hag got close to deal-making HP, I made sure it was Karlach who was active, gave the hag one last hit to achieve the correct HP-level, then ended Karlach's turn. I don't recall if I had to end any other companion's turns or if it's enough for the hag to have the correct HP-level and your currently selected character to end their turn.
Once the hag was up though, the dialogue triggered and Karlach was the one doing the talking.
Even if oath-breaking isn't an issue, if your Tav isn't good at intimidation, it's always a good idea to bring someone who is. I *think* intimidation is the only way to make her both drop the hair and set the girl free.
S.