You can combine clawhammer and fingerpicking but you have to compromise a bit for each style. Use a thumbpick and and fingerpick on your index finger. Do your clawhammer playing with your middle finger performing the downstroke and thumbpick catching the drone string. You will have to practice to keep from hitting the thumbpick against the head.

Clawhammer can also be done with the index fingerpick performing an upstroke, then catching the first string as an upstroke, followed by the thumbpick hitting the drone string.

The two methods can be combined and distinct features of clawhammer (brush stroke) and distrinct features of fingerpicking (pinch and upward note sequences) can be implemented.