I think you will need longer than 3 nights in Beijing. This organization might be able to help you find a specialty expert guide suitable for your specific interests. Contact 2-3 months in advance.
http://www.chinaculturecenter.org/

May is good weather time most everywhere in China. There is a national holiday May 1-3, and many people choose to extend this period and travel around. If your timing is after May 8, no need to worry.

Might be worth putting a couple of days/nights in Xi'an between HGH and Beijing. Use flights. Quite a bit beyond the Warriors to see there. Research and see. IMO, I think 3 nights in Hangzhou, even at an Aman, might be 1 night too many.