Yeah, if using the Nano with only a 3.7v, definitely don't use the LEDengin. The greens will want right at the battery voltage to run well, and with what the board needs for the speaker etc, you won't get optimum brightness out of it.
Go with a Tri-Rebel GGG if you want the MOST brightness, all run in parallel that's 3A but I think the Nano will be limited to 2A which will still be brighter on those three dice than any other LED. That said, if you want 'good enough' and bright, go with a single 'highest lumen' Rebel green or a Seoul P4.