When connected as a current regulator the dropout voltage has an extra 1.25V added to it.
You're right it won't be 4.25V but it'll still be too high for what you're doing. To meet the specifications on the datasheet the voltage differential should be 3V, at lower currents it will be lower and don't expect the ripple rejection, regulation etc. to be as good as specified.
Suppose the dropout voltage is 1.8V, you need to add 1.25V across the sense resistor giving a dropout of 3.05V. Three LEDs in series have a forward voltage of 10.5V so the regulator will need 10.5 + 3.05 = 13.55V to regulate properly. Once the input voltage drops below 13.55V, the LEDs will start to dim.
I would opt for a switching regulator anyway because it's much more efficient.