Hi John,
The peak current is a little hard to set a definitive number on. The transistor can handle 40 amps but it has a 0.025 ohm resistance fully turned on which means you would get a 0.5V drop across the resistor at 20 amps. So, if you were down to 7.0V on your lithium batteries and took into account the 0.44 volt drop across the BatShare in the SuperReg then you could pull 20 amps without causing the regulator to drop under a regulated voltage of 6.0V. If you did not care if the output dropped below 6.0 volts some then you could do more than 20 amps.
I hope this helped. Let me know.