New Jersey Copper From Junk Tray

Why?

I see no reason to have it slabbed. Some different pictures may help attribute the variety. You can also try making a rubbing with aluminum foil, I have often done this on lower grade coins, it helps to see some of the details.

Although this piece is a significant part of early American numismatic history, it is in such poor condition as to not make it worthwhile to have it authenticated.
A better approach would be to take it to another third party, (such as us) for an opinion, or to a respected coin dealer. Certainly cheaper than having it slabbed.

Even it it were slabbed, the opinion of the slabbing third party is, after all, only another opinion.

Another reality that has to be faced, is that genuine or not, is that huge amount of deterioration that this coin has, can cover up all sorts of evidence that would otherwise prove it to be genuine or otherwise.
That makes it almost impossible for anyone to provide a completely unchallengeable opinion, including a slabbing third party.

By the way, I would have also snapped it up out of a dealer's junk bin at the price you paid for it! Well done!