I am assuming as you show searching for "ObjectSid", you are using Microsoft Active Directory. If so, You can to use the extensible matching filter using LDAP_MATCHING_RULE_IN_CHAIN. Query the group with the scope=BASE and Something like:
(member:1.2.840.113556.1.4.1941:=CN=John Smith,DC=MyDomain,DC=NET)
Should work.