MOSFET as Power Switch Issues
If I understand, the problem is R2. When you apply a load, you begin to draw current from the battery and it must flow through R2. According to V=IR, this will create a voltage drop across R2. So, at 0A, the voltage at BAT_SW is 3.7v. At 10mA, BAT_SW=3.7-300*0.01=0.7v. So the boost converter is struggling to boost 0.7v up to 5v and the problem gets worse as it tries to draw more current. 

The real problem is the configuration of the transistor. You are turning off the boost converter by diverting power around it to ground. So even when it is off, you have current flowing through R2 to ground, effectively burning (3.7^2)/300=0.045W continuously.

I recommend you research “mosfet as switch” to find some tutorials about this. You probably want to use a P-channel instead because N-channel configurations disconnect the switched circuit from GND so you really have 2 grounds. This is fine for motors and stuff but probably not boost converters.