Can you describe the problem rather than stating what you suspect the problem is.
There is enough fuel in the carbs to let the engine run for a short time, I'd guess 39s to 2 mins.
So even if the fuel pump was not working, I would expect the engine to start.
I may well be wrong but I don't buy the theory that the fuel pump does not run yet the battery is able to turn the engine over.
When you turn on ignition, do you hear the pump click a couple of times until the carbs are full?
A way to check is undo the hose from the fuel pump and see if it pumps.
When the engine started after charging the battery, how long did you leave it on the charger?
Also if you did not charge the battery after a run, does the engine start from cold?
How old is the battery and have you measured the voltage across the battery when engine is not running and when running at about 2000 rpm.