Greetings aspirant.

For admission in BAMS private colleges, you will have to apply at the official Ayush site of that particular state. Private colleges do not come under the All India Counselling. The registration fee which is non refundable is 500 rs for government colleges and 5000 for deemed universities. You will also have to pay refundable security of 10,000 rs for government colleges and 50000 rs for deemed colleges. After this, select the colleges you want to apply in. For your convenience, I am linking the site below-

https://aaccc.gov.in/#/home

Hope it helps!