What’s the best way to determine which version of Oracle client I’m running?

You can use the v$session_connect_info view against the current session ID (SID from the USERENV namespace in SYS_CONTEXT).

e.g.

SELECT
  DISTINCT
  s.client_version
FROM
  v$session_connect_info s
WHERE
  s.sid = SYS_CONTEXT('USERENV', 'SID');