You can forego faa medical if you have a current flight physical and are only exercising third class medical.
14 CFR 61.23(b) Operations not requiring a medical certificate. A person is not required to hold a medical certificate—
(9) When a military pilot of the U.S. Armed Forces can show evidence of an up-to-date medical examination authorizing pilot flight status issued by the U.S. Armed Forces and—
(i) The flight does not require higher than a third-class medical certificate; and
(ii) The flight conducted is a domestic flight operation within U.S. airspace.
Edit-a lot of military flight docs are authorized to do no-joke FAA second and third class medical certificates as well. Ask your local flight surgeon for details.