Well I'm sure you will continue to get ideas, but for me in that case, we wouldn't be done lunging until she did it. And if a lunge whip keeps her moving, so be it. She seems to think it's an option to call it done? Not fatal but stubborn and the more times she ends the lunge session by stopping or not moving, the more she'll do it. I think until you see her reaction when you push her, being at a distance with a lunge whip is a good idea.
I don't know about others, I haven't had to hit a horse with a whip before - crop yes (but that's only for close work) but not the lunge. The ground behind him is enough to mean business. Or in front when I've cued him to stop and he ignores me....only takes once or twice.
Like I tell my kids, he just wants you to PROVE you can make him listen. Once you do, it makes a huge difference. She wants proof :)