OH, I don't think it's silly to say that a crop will ruin your connection. She may be resentful and worried about it. But it might be what is required for her to be honest about keeping a gait.
Is there any physical reason why she cannot canter any great distance?
Are you , by any chance, inavertantly giving her a signal to slow down, like being too rigid in your body? Or is it just school horse laziness?
She is a school horse, no? It's hard for them to continue to be mentally engaged in their work when it's so much the same , day in and day out.
You might refresh her by taking her for a long hack and letting her really gallop up a big hill, maybe "race" with another horse.
YOu can also take you hand and give your own thigh a good, loud SMACK and see if this doesnt reinforce your leg, . I mean ask for GO! And if she gives you go (tiny go), then smack your thigh and literally say,"I mean GO!".
Raise the energy in YOU to raise the energy in her.
I do appreciate that you ride with sensitivity to this mare's personality.