Charging a battery with LED on at the same time?

I could use some help. I have a design where a rechargeable battery and LED are connected (think flashlight) and I'd like to charge the battery while the LED is still on in a dock. The part I'm purchasing has the battery and LED directly attached to each other, so I can not separate out the battery from the LED.

I'm using a MS920SE battery - data sheet here: (https://www.sii.co.jp/en/me/files/2024/01/MicroBattery_E_20230330_rev05-security.pdf) The data sheet recommends constant voltage with a current limiting resistor in the charging circuit.

I've drawn the schematic here:

(note the voltage regulator is not shown, but would S-13R1 per the datasheet)

Based on testing, the light draws about 14 mA at 3.1V and the battery charging circuit should be limited to 5mA at 3.1 V.

My Take:
In order to ensure the battery remains at a constant current charge, I need to detect if the light is on or off (current sensor over a resistor to microprocessor). To sense if the light is on, I would use a transistor to add a resistor in parallel to enable ~19 mA to flow. If I sense the current at a lower current (<5mA), or higher current (~20mA+, potential short) I would remove the voltage to the transistor and limit the flow to 5mA. I'm also considering current-limiting diodes for this application instead of resistors.

Any recommendations?

(Side note, I would also appreciate an explanation as to why adding the LED seems to make the circuit act as a voltage divider. In the simulations, this seems to limit the voltage that the battery will charge to when the LED is on.)

It IS a voltage divider.

Use something like this instead, which does not affect the battery charging current (for ideal charging voltage source V1).

Capture

R1 limits the battery charging current, R2 limits the LED current.

I’m not able to change the led & battery parts of the design as I’m buying them COTS commercial off the shelf.

The battery is connected to the LED - I need a solution where those elements are connected.

Can you post the complete schematic of just the flashlight?

I don’t have it, but it’s pretty simple. An LED, a transistor, and a magnetic Hall effect sensor for on-off.

I’m leaning towards the current limiting diodes at the moment. Just limit the baseline current to charge the battery and open up the additional current if I detect the LED draw.

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