Try the phone or email contacts at Florida Department of Law Enforcement - Home.
The state of Florida doesn't appear to maintain a statewide officer registry, but that department should be able to verify someone is or isn't a officer in conjunction with the police department closest to your location.
If this neighbor is threatening or attempting to enforce the law while not showing ID that proves he/she is an officer, that's serious, and you should call the local police department.
Talking about being a LEO while drinking beer at a BBQ is less serious, but still could be a third degree felony.
See Statutes & Constitution - Florida State Statutes:
843.08 False personation.—A person who falsely assumes or pretends to be a firefighter, sheriff, officer of the Florida Highway Patrol,
officer of the Fish and Wildlife Conservation Commission, fire or
arson investigator of the Department of Financial Services, officer of
the Department of Financial Services, officer of the Department of
Corrections, correctional probation officer, deputy sheriff, state
attorney or assistant state attorney, statewide prosecutor or
assistant statewide prosecutor, state attorney investigator, coroner,
police officer, lottery special agent or lottery investigator,
beverage enforcement agent, or watchman, or any member of the Florida
Commission on Offender Review and any administrative aide or
supervisor employed by the commission, or any personnel or
representative of the Department of Law Enforcement, or a federal law
enforcement officer as defined in s. 901.1505, and takes upon himself
or herself to act as such, or to require any other person to aid or
assist him or her in a matter pertaining to the duty of any such
officer, commits a felony of the third degree,