Charging a li po battery while using it
The solution is to power the circuit from the solar panel when that is available while simultaneously charging the battery.

Obviously you can't simultaneously charge and discharge the battery. Current is either flowing in or out - not both. Drawing power from the charging circuit at the battery terminals will confuse the charger as it will assume that all charge is going into the battery.

Ali Express is not a good source for components because you usually can't get proper datasheets for the devices. That is one of the reasons the products are cheap. The moto here is, "No datasheet - no sale".