NPN transistor switch wireless charger module
The wireless charger module you referenced (https://www.dfrobot.com/product-1283.html) is based on a switched capacitor voltage regulator on the receiver side.
This regulator has a &quot;&quot;skip mode&quot; which is an automatic shut down if the current drops below 4mA. https://datasheet4u.com/datasheet-pdf/WillSEMI/WD3168/pdf.php?id=1112865
It is quite possible that the V-CE voltage drop across the transistor reduces the charging current to below 4mA when the battery is charged somewhat but not fully; at which point the regulator is shut down.
If your design is married to this regulator/charger, one work-around might be to artificially increase the current drawn from WC+ by shunting a resistor that guarantees a current draw of at least 4mA. This would slow down the charging a bit during the charge phase but keep the charging going for longer, and on balance provide a faster charge.
For a supply of 4V and a current of 4mA, a 1KOhm across WC+ and GND might be just what you need.