For India, our normal TA uses an India-specific agency, but then Aman themselves were pretty active in the planning when we told them what we wanted to do.
If you can make it work, I would spend 2 nights at Aman-i-khas, three nights at Amanbagh and then 1 night in Agra. Bagh-to-Agra is a decent drive and it is nice to be there in the morning for the sunrise which you would miss.