Great video thanks. Very interesting. In many ways.
If you can engage her via lunging, then one of the real benefits will be to teach her to engage her hind, and halt and half halt by bringing her hind underneath. You can see at the moment she has issues in the stop. It's stiff, stuttery and forehand loaded though she is actually trying to find another way, you can see that. She just needs a little guidance.
That's what I would be doing with her. However that's a very physical type of lunging in the sense that you have to walk the circle actively and engage your own body, it's not something you can do in the 'turn on the spot' type of lunging. So it depends on heat you are happy to try.
Either way, the important thing with a horse like this is to ask for a few steps only, then a stop. A few steps more then a stop. It's down to feel but there are two reactions from a horse like this. If they are nervous they will be trying to turn their head away from you. Being happy to turn in is not a bad sign. If you try to force them out with too much energy they will likely take off. Remember you want then to be happy to turn in because then you can ask for just the front for shoulder in etc later. If you get them to bring their hind under to stop then all of a sudden they cannot turn in until after they have stopped, and as a separate action which is ok.
I would definitely suggest using some sort of whip, only so that you can be precise as to where you are pointing.
Will she lunge in a straight line rather than the circle?