How exactly do you reward her for staying out with a treat? You would have to bring her back into you or go out to her, which defeats the purpose! Definitely don't use treats to teach lunging, I've never heard of such an odd method :/
If she's not worried about the whip, then you'll have to use your body to back it up. A mare wouldn't take it if the filly kept coming into her space when she wanted it out would she? So be the mare, if the filly makes a move to come into you, ask nicely first by pointing/flicking the whip towards her girth/shoulder. And if you get no reaction, then move defiantly towards her and growl at her, wave your hands if you need to. She might jump back, but that's fine, just get her out of your space.
It's fine to desensitise horses, but they need to have a small amount of fear and respect for you and things like whips. Otherwise you create you may actually be creating a monster. Make sure she knows that you **** well mean it when she comes in on you. And stop the join up at least until she's really got the hand of lunging. They are two very different things but to the horse, you're running it on a circle and normally you want it to come in, so why isn't that acceptable now? Make it known that you want her to stay out from you and ease up with join up.
G&K'sMom had a good tip, having someone walk with her. You can start with them leading her out onto the larger circle, then removing their hand from the halter so she's walking on her own, and slowly backing out towards the middle of the circle, it is just about the same practice as what I described above, but with 2 people.