I am a not an FAA licensing expert, however,
When I look at the "ratings" available for issue from the FAA as listed in FAR part 61, I find the following:
(b) The following ratings are placed on a pilot certificate (other than student pilot) when an applicant satisfactorily accomplishes the training and certification requirements for the rating sought:
(1) Aircraft category ratings--
(i) Airplane.
(ii) Rotorcraft.
(iii) Glider.
(iv) Lighter-than-air.
(v) Powered-lift.
(vi) Powered parachute.
(vii) Weight-shift-control aircraft.
(2) Airplane class ratings--
(i) Single-engine land.
(ii) Multiengine land.
(iii) Single-engine sea.
(iv) Multiengine sea.
........
(7) Aircraft type ratings--
(i) Large aircraft other than lighter-than-air.
(ii) Turbojet-powered airplanes.
(iii) Other aircraft type ratings specified by the Administrator through the aircraft type certification procedures.
[(iv) Second-in-command pilot type rating for aircraft that is certificated for operations with a minimum crew of at least two pilots.]
(8) Instrument ratings (on private and commercial pilot certificates only)--
(i) Instrument -- Airplane
(ii) Instrument -- Helicopter
(iii) Instrument--Powered-lift.
I don't see anything specifically referring to a constant speed prop, flap or retractable gear, or even "complex" specified in that list. Thus, I would not expect to see "complex" as an endorsement or rating issued to an FAA license. If you have a license whis says "Airplane" and you're flying one within the "class" endorsed on your license, and it does not require a type endorsement (and small "complex" single engine propeller types would not), I would think that you've met the FAA licensing requirements.
It is likely that the owner of the aircraft will want you take some training on the plane (though, if you've read and follow the flight manual, constant speed props are pretty simple), but other than that, I cannot see that the FAA cares one way or the other.