I tested the CC BEC a while back. The bottom part of the results below is what you're interested in.
As you can see, I had it set for 6v. If the input is below 6v, the output tracks pretty close to ~0.3v lower than the input (looking at the higher current 2.5 ohm resistor load column). At around 4.5v, the BEC output shut off. I think you'll find that most step-down BECs will have similar results (if not worse). So, an LVC of anything below ~5v results in unpredictable behavior, and that's not including when the voltage dips for brief intervals on high current bursts.
To avoid all this, you could get
Dimension Engineering's AnyVolt 3. It's quite expensive, but it will provide your set voltage no matter if the input is below or above your set BEC output.
EDIT: Nevermind about this unit because it needs an input at least 5v.