I would buy a CC3D and get it coming in the post if flying is more important than the $15 or so for an exact replacement. Non-EVO (does not have DSM receiver connections?), full sized CC3Ds are $10 shipped on eBay.
Plug in the CC3D with the battery and no USB and let it sit for 10 minutes. Does anything get hot anywhere? ESC, component on the CC3D? One ESC might get a little warm, but not hot. If any tiny component on CC3D gets surprisingly hot, it may indicate a problem area.
I would try one more thing and then buy a new CC3D if that doesn't fix it.
Plug in USB and do File -> Export UAV Settings. This saves your current settings to a file. Save the file in a place you can find it.
Then do System -> Halt
and Erase Settings when it becomes active after Halt. Follow instructions.
Then test if there is a difference in the LED when booting from battery vs. booting from USB.
Finally, do File ->
Import UAV Settings to put your settings back the way they were and see if it works.
Keep this file. You can import it into a new CC3D, do sensor calibrations, and fly, rather than doing full setup.
Chances are that there is crash damage on the CC3D that causes this. This particular issue sounds like it is caused by bad connection or component going from ESC/servo connector to internal 3.3V voltage regulator. It could be fixed if you can read a schematic and do very tiny soldering, but buying a CC3D is a lot easier.