I've been reading this entire story. You have a transfer student who I imagine has been going along this kind of path, with parents acting as (uninformed?) advisors all along. So now you have someone who does things well which come naturally, while missing a pile of basic skills. You already know that he is not ready for this event that the parents want him to do. From what I gather, the student has neither the technical skills, nor the reading skills. I don't understand why you are even considering this.
No, it does not make sense for you to be charging more. A professional should be doing his best at all times anyway, and you can't be doing more than your best. The extra practice time is up to your student. The only thing that I would see is more lessons or longer lessons, in which case more hours equals more pay. But tbh, I question the decision of going ahead with this at all. In fact, is it a decision, or just going along with the parents?