Have you tried a 1.5V non-rechargeable cell? Your rechargeable cell may not provide sufficient voltage to properly operate the lights.
I note your initial post indicated a cell voltage more akin to a lithium cell or 3 x 1.5V cells as a battery. Which is it?
Is there any provision for an alternative power supply?
With the correct voltage, the momentary button likely changes the mode of operation (on continuously, short or long flashing plus possibly some others).