Can I get cosmetic surgery (rhinoplasty, jaw surgery, maybe breast augmentation) having Chiari malformation?

Plastic surgeons while aware of chiari malformation certainly are not experts on this. Therefore, prior to having surgery, you would need to get clearance from your neurologist or neurosurgeon as well as meet with the anesthesia team prior to undergoing surgery.

There is no reason you can not have any of the surgeries you mentioned having a known Chiari malformation. Even in otherwise healthy patients with VP shunts they can undergo any form of aesthete face or body surgery.