5V DC power to recharge 3.7V LiPo and power a circuit at the same time
You could charge the battery with a wireless tickle charge to extend the operating time or sustain it.  A voltage protection to cut off charging current of 3.8V is required when the LEDs are off with a simple float charger.
Wireless can use the stripped conductors as a loop inductor with a reverse diode. Transmitter could be a 5 to 10W amp at 100k to 250kHz with tuning required using  metal film caps for resonant coupling.  If the secondary LED wiring fails to give the desired results, duplicate the sender wiring and use impedance calculations to choose the right values.  Proximity should be less than the smallest loop diameter.  The ratio of diameters could be 1:1 or use the same length of wire for many loops in each.  LED resistance will be around 15 Ohms each so inductive reactance ought to be less than 15/30 ohms
Use the impedance RLC nomograph in my recent answers.
This is just a paper making design concept.
Recommendation
The simple solution is drop 5Vdc to 3.7 to keep the batteries going using 2 power diodes (10A) or a power LDO like LM317 with a sink.