If Amps is Equal to Voltage divided by Resistance, which I think it is, then you are drawing about 2 Amps when you Smash Down the Button.
Actually a Little More than 2 Amps because No PV/Battery is 100% Efficient. That Really isn't a Lot of Amps.
If your AW 18490 is 1100mah. That is the Same as Saying that it is 1.1 Amp Hours.
So in a perfect world, that battery would be able to work for 1 Hour Straight when it is being asked to supply 1.1 Amps. But you asking it to do more. You are asking the Battery to Supply 2 Amps.
So your Battery should run for about 30 Minutes Straight at 2 Amps.
Since you say that you Like to take Long Hits, 4 Hours of Long Hits and hitting it Pretty Regularly sounds about Right.
dang, you're good
well, at least I know it's right
Now I need 18650 batts and the extension cap