Before you mentioned the battery was 3.7v Now you say it’s 5.7v
The battery is 800mA-hr Not 800ma.
You cannot test the circuit I designed on a simulator as it does not take the impedance of the panel into account.
The output of the circuit will “magically adjust” to the voltage of your battery, so you don’t need any zener diodes.
As I said before, when the voltage of the panel reduces in voltage the current decreases enormously.
Internal lighting has “no energy” and the panel will not produce a worthwhile output.
Just put a diode on the panel and it will charge the battery when sunlight hits the panel.
The whole thing is much more complex than you realise and you are going about it entirely the wrong way.