How to get TNS connect strings (like tnsping) to check a dblink via a session?
I do not think that is possible. 
Someone or something needs access to the Filesystem to read the tnsnames.ora.
When using the DB-link oracle will get the definition for RDBX from the tnsnames.ora file. This is transparent to the user, so the user will not notice. As @EdStevens suggested I would try to solve the political issues and work with the DBA.
Or, ask the OS-admin to have a look for you.