I just read the rest of this forum, and here's a possible idea for the amp circuit (again, its cutting off a bit on the right, that's just a Ds for where the leds are):
So the + and - on the left are the + and - at the right of my last picture. - is tied straight to ground and + goes to the negative input of an op amp. The positive input on the amp is connected thru R0 to ground. The amp is powered by the voltage on the lines. There is a negative feedback connected thru RF from the output to the negative input. Ok, the ratio R0:RF determines the amount of gain, you want this to be high gain, I don't remember exactly how this works, if high gain is from high R0:RF or low. This high gain forces the signal to the rails, so that the output is equal to the power on the power inputs. As I'm typing this, it seems to me like this circuit doesn't actually do anything, we're only getting out what we put in. So what it really needs is an independent power source that is slightly more power than what you want on the output. This way, if you have the high power of both supply and button from the last picture, the gain from the amp rails the circuit so that it is fully on. When you have only the power of the button, the gain is enough so that on the peak of the pulse the signal barely makes it to the rails, and then falls off as it loses power.
I'm not really sure where you can get that independent power, but I may have an idea that could work. Let me know if this much makes sense to you and if I can clear anything up.