Hi, Felix,
Would you like to use a processor to do the job? If you want to use a processor, I think you can use for example LPC804, you can connect GPIO to LED via a serial resistor. The GPIO source current can reach to to 20mA with 3.3V output. It has ADC, you can connect the battery voltage to ADC, when ADC detects the rising voltage, drive LED to on.
This is LPC804 link:
LPC804M101JDH24 Product Information|NXP
Hope it can help you
BR
Xiangjun rong