I sort of have a solution, maybe ... if you can use a multimeter ... When your PC is shutdown Find a +5v that's live on the 24 pin ATX power supply. You could then pop the connector out along with a ground and solder into it/splice/whatever.
Then you can put the power button onto the momentary ...
HOWEVER ... If the +5v to light the LED is also connected to the momentary, you will pop something. What you might in actual fact have to fit will be a powered relay switch to make a break for the momentary to be enabled/disabled. It will be an NO relay.
You will want to buy a project box for it if you aren't modding it into your case, but Id use testboard and hook it up.
Make sense?
You put the relay switch in parallel with the LED. Hook up -ve directly to memoentary, run wire from +ve to one side of the NO gate on the relay, then the other side to the momentary.
Now when the LED is sent power, the relay closes and allows for the momentary button to be pressed. I know for a fact it works, I've done it.