Among the three you have mentioned, Homeopathy is the course for which you can go , if  you wish to be a clinical medical person after your UG course. The clinical exposure is better in Homeopathy compared to that of Ayurveda. Ayurveda is also good but most of the employment is for the ayurvedic manufacturing firms. So both the course are good but you have to choose the field which matches the best with your target.

The score need for the course of BHMS is around 450-550 & for BAMS is around the same.

For course of BHMS you can visit the link below:

https://www.careers360.com/courses/bhms-bachelor-of-homeopathic-medicine-and-surgery

For course of BAMS you can visit the link below:

https://www.careers360.com/courses/bams-bachelor-of-ayurvedic-medicine-and-surgery

For Government BHMS colleges you can visit the link below:

https://medicine.careers360.com/colleges/list-of-medical-colleges-in-india?ownership=Public/Government&degree=B.H.M.S.

For Government BAMS colleges you can visit the link below:

https://medicine.careers360.com/colleges/list-of-medical-colleges-in-india?ownership=Public/Government&degree=B.A.M.S.

I hope this helps. Wish you a great eve!