Most of the beaches I see on websites and travel guides are too big and touristy to my taste. Are there any small beaches near Sanya (max. 1.5 hours by bus).

Haitang Bay is suggested to visit. From the local General Bus Station, you can find a bus to Haitang Bay. The buses are operated from 06:50 to around 20:00 and the bus duration is around one hour to Haitang Bay.
I am sure you will find the beach you want to see there.