Aunty's advice in this case will depend on your primary goal. 

If you mainly want to be a fireman, then tell the third party polygraph company that they already have test results for you on file. They will decide whether to re-test you or use one test for both clients. Do your best on any second test and don't worry, they won't let you fail. 

If you mainly want to prove that the polygraph lie detector test is hokum, don't tell anybody you have already been tested. Do your best on the second test, and raise a big stink if the two test results are in conflict. 

If you mainly want to be maximally entertaining, don't tell anybody you have already been tested, and deliberately fail the second test. 

Aunty would pay admission and buy popcorn to witness the civil lawsuit over the tort created by this third party polygraph company when they test the same person on the same questions for the same purpose, and get two conflicting results, one of which has damaging consequences for the testee. 

"Your honor, my client was denied employment by <fire department> due to the 'deceptive' rating given my client by <third party polygraph company> on <date>. This rating is demonstrated and proved to be negligently faulty by the 'truthful' rating given my client by <third party polygraph company> on <date>." 

In any case, good luck, and tell us how it comes out.