Is it possible to enter a missing sequence number into another table, on exception in a Trigger
EXCEPTION 
  WHEN OTHERS THEN 
        SEQ_AUDIT_PKG.Inse...


is a bad idea, because 'when others' could be anything. eg, you might have a datafile corruption, or an out of memory error.  You dont want to be doing a sequence audit in that respect.  Similarly, when that primary trigger fails, you want to be sending that back to the calling program, not telling them that "Hey, all is well" (which is when WHEN-OTHERS typically will do).

You would need your SEQ_AUDIT_PKG.InsertMissingSequence to be an autonomous transaction so that you could log the information outside the transaction scope of the trigger.

But having said all that ..... just dont do it :-)

Sequences have gaps...period.  You can


nocache them
pin them in the shared pool
carefully police shutdown abort commands
capture unused ones with an autonomous transaction


and you'll still eventually get some missing and any promises you've made about no gaps become moot.  And the complexity cost toward meeting this goal is just not worth it.