Christian, neither Edward Callahan's volume, LIST OF OFFICERS OF THE U.S.N., nor the general index volume of the OFFICIAL RECORDS OF THE UNION AND CONFEDERATE NAVIES IN THE WAR OF THE REBELLION, nor the 1865 NAVY REGISTER have the name of a John Perry Willets listed. However, as with all such sources, there may have been errors, and perhaps Willets did serve in some capacity.
As you indicated, the National Archives is probably your best bet. I would also suggest contacting the Naval Historical Center about anything on Willets.
Good luck in your research.