You will have to rent a car and drive down to Punta Allen. The flats are not within wading distance if you want to wade the flats. However, you can walk up and down the beach fish from the beach all you want. I have done this and it can be successful. (Read my Ascension Bay fishing report)located in this forum.
I don't know of the "American Place" to stay that was mentioned so I can't reply to this. The whole area is very low key as you say.
Have a great trip and hope to hear how you did.
PS: I forgot to mention---GO STEELERS!! I saw you were from Pittsburgh, I hope your a Steeler fan