PNP Transistor Not Working as Expected with Higher Voltage
Your issue is that you have Q1 configured as an emitter follower.  So the voltage on the emitter will be about 0.7V higher than the voltage on the base.
With your 12V adapter, when the charge ends you didn't have enough voltage across the diode to turn it on.
With the 15V adapter you'll have about 10.5V on the base when charging is done, and 11.2V on the emitter.  If your diode drop is say 2V you will have 15-2-11.2 = 1.8V across R1, and 1.8mA in the diode.  Probably enough that it still glows.
No easy solution by changing the transistor, you'll have the same result.  It has nothing to do with breakdown.
If you don't want to change your PCB you could try a bigger resistor for R1.  Increase it until the LED no longer glows and see if you get acceptable brightness during charging.
The better approach would be to use the transistor as a switch, but that's a PCB layout change or lots of cuts and jumpers.  JRE's suggestion of putting some LEDs in series would also work.