State doctors regulated - yes, private - sky is your limit
What regulates charges are insurances - well, I won't talk about state... normal privates - up to x3.5 of the standard charge, super duper privates usually no limit, however its really hard to afford them if you have family and kids (if you are normal person that is
)
I may also go for a second opinion to Alphaklinik if don't see any improvement, however from my experience patients with private insurance way too often are treated like a sack of money...