As Adam says, it’s called tmin. I set it by script all the time, using yyyy-mm-dd date format.
A fragment from my tmin script
curl -# -X PUT -d "{\"tmin\":\"$TMIN\"}" https://www.beeminder.com/api/v1/users/$username/goals/$GOAL.json?auth_token=$auth_token --header "Content-Type: application/json"