Constant current driver and undervoltage protection for lipos
Most small LiPo cells sold for electronic hobbyists (not RC flyers) have a built-in protection circuit like this one from Adafruit. It's a small PCB under that yellow mylar tape at the end of the battery. It already has an undervoltage cutoff function to protect the battery, so you don't need to duplicate that function. It does not oscillate, and only reactivates when a charging voltage (well above the cutoff voltage) is applied to the wires.  You should inform DIY builders of your design to only use LiPo's that include built-in protection.
With that out of the way, you can use any LED driver that meets your specs.