Your button shows four important ID-characteristics:
1- Its anchor lacks the "fouled" rope usually seen on US Navy buttons.
2- The anchor's top-bar is curved.
3- The anchor is connected to the inner ring by two loops, not just one loop.
4- It SEEMS to not be made of pewter, but instead looks like brass or bronze.
Those characteristics cause me to think it is a 1790s French Navy Officer button. See the button on the right in the diagram below, from a very-extensive book on French Military buttons (titled "The French Uniform Button) . That button is reported to be made of bronze. The book is viewable online, but it written in French, so you'll need a translator-program to read it. To view the diagram (and info), scroll about 3/4ths of the way down the webpage, here:
http://detektorysci.pl/guziki-napoleonskie.html#bookmark293
It leads me to suspect that the curved-top, no-fouled-rope buttons shown in the Albert book as button NA-2 is an American cast-pewter copy of the bronze French Navy Officer button.
The next buttons in the Albert book, NA-3A and 3B, are listed as "French style" US Navy buttons. However, the exact-same NA-3A button (which is made of brass/bronze) is shown in the "The French Uniform Button" book's section on "Military Naval and Colonial" buttons as being an "Old Regime" French Navy button, dating from 1772 to 1789.