Driving P-Channel MOSFET from MCU Pin While Running on Coin Battery
I think it's nothing to do with the MOSFET- the CR2477 simply has too much internal resistance to drive enough current to run your circuit. As the battery terminal voltage drops, your boost regulator will try to draw more and more current to keep the output voltage stable and at some point cannot work. According to the datasheet, the voltage is starting to drop substantially even at 1mA.  



That is typical for lithium coin cells.