How to calculate ripple/smoothing cap for vintage motorcycle LED headlight conversion
It sounds like what you want is a LED power supply that has a wide input voltage range and drives the LEDs with a constant current, so long as that amount of power is available.  

Since the input voltage can be 4-16 V AC, you probably want to use a boost converter.  A buck wouldn't leave much room with 4 V input.

First full-wave rectify the AC to make DC.  The peaks will be the square root of 2 higher than the AC voltage, minus two diode drops.  At this low voltage, you can use Schottky diodes, which drop less voltage and full silicon diodes.  To be pessimistic, assume the the Schottky full wave bridge drops 1 V.  That leaves the peak voltage coming out of the bridge to be 4.7 to 21.6 V.  Put a nice a big capacitor on that, and you have a rough DC power supply.

String enough LEDs in series so that the total needs more than 22 V to light up, then have a boost converter power the LED string from the 5-22 V input power.  Instead of regulating the voltage to the LED string, have it regulate the current thru the string.  To get a voltage proportional to the LED current, put a small current sense resistor in series between the bottom end of the LED string and ground.  For example, if the LEDs are meant to take 200 mA, then a 2.7 &Omega; resistor will drop 540 mV when the LEDs are running at the intended current.