Solar powered LED light needs help
Assuming 12h LED duration, current is ~156mA (0.5W/3.2V)
thus 1900 mAh is needed (156mA*12h) 
you are getting a few hours with a 1000 mA LiPo battery charged only to 3.2V 
thus not getting full capacity of up to 6h (minus series losses from 3.7 to 3.2 or ~14% (0.5/3.7)


If 16Vac Tfmr is unloaded, it is reading 14% higher due rated V at 86% efficiency.
the transformer can supply &lt;2A @30VA @16Vac only to a linear load and not to a bridge rectified battery but a series R  can make it more linear yet lossy with &lt; 15% efficiency 3.5/24V in powering the LED.
thus another wire pair from a full bridge diode at transformer or single wire to bridge and series R at load must be wired to charge the battery using current load for expected voltage R drop.
Using 16Vdc avg to 3.2Vdc LEDs with up to 156mA 
with no PV, R=82 Ω drawing 2W (12.5*0.156A)
however when LED is off, Vbat continues to charge up during day time with no overvoltage protection.  



Thus you can leave LED on all the time, and choose a 5W resistor for cooler operation or buy a proper supply or add more complexity to regulate LiPo charger.

Unfortunately both your PV and storage battery are under-designed to meet the load requirements.


  One simple solution is power the battery and LED from a 3.3V source such as tap into an ATX PSU with wire pair to series Sch. diode to LiPo, or buy one ~3.2Vdc >150mA wall supply to feed circuit.