Yes she will cross fire on the lunge but not as much as in the round pen.
I haven't used the round pen in a while because the circles make it worse due to lack of balance, however I have been hesitant to canter her at all in any situation to be honest.
I did ask her for some short periods of canter in both directions on the lunge when I had her in side reins (one fixed on the outside and the lunge line through the bit and to the surcingle on the inside) so that I could flex her and then release if needed. She seems to do better with this set up than with two fixed side reins. She maintained the canter longer, but still cross fired as she picked up speed.
It's hard to explain but I feel as if she "psyches" herself out when she's cross firing. I'm assuming it can't be comfortable to have half your body going one way and half of it the other way, but it is hard to slow her down as she accelerates her speed.
I have not been able to ride her outside, say in a straight line or through an open field, due to the snow and mud
Sorry for all the confusing explanations...I have been trying to do my research but I realize not every horse is going to be the same. I'd like more opinions before I start trying to work on this problem, especially if it ends up being the wrong approach.