I don't think that is good idea connect LED in buttons place on Shift Register or Diode Matrix*, connect to external power source, e.g. USB cable + and - wires. *

And use 2 pins from DPDT switch for turn LED on together with button on MMjoy2. Can use a DPST (ON-OFF) for this.

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MMJoy2 LED support is for output LED and work for War Thunder, not all games.

* But if want connect there look back in this topic someone post how he connect LED in between diode matrix.