I suspect that you are trying to draw well over 5A, which is why the output voltage falls. The maximum voltage across the 0.3Ω load is:
$$ V = IR = 5A \times 0.3\Omega = 1.5V $$
This is suspiciously close to the 1.3V you have witnessed, if the MOSFET were switched fully on.
You must apply exactly \$V=IR=5A \times 10m\Omega = 50mV\$ across the 10mΩ sense resistance, which will be hard to get right, especially if the op-amp has millivolts of input offset. Any potential over 50mV at the op-amp's non-inverting input will cause this over-current condition. In fact, depending on the op-amp, this might not even be possible. A TL081, for instance, won't work with inputs closer than 1V to its negative supply.
On top of that, after the heater has eaten 1.5V, most of the the remaining 0.5V is across the transistor, which will therefore be dissipating \$P=IV=5A \times 0.5V=2.5W\$, which seems a waste. It's 33% of the heater's power.
It makes no sense to use both PWM and the voltage controlled current sink that you have implemented here. Use either one or the other. I presume you are trying to limit current to 5A, given a voltage source of 2V, but this is better achieved with PWM from the correct voltage to start with, like this:
simulate this circuit – Schematic created using CircuitLab
This ensures minimum power dissipation in the MOSFET, will switch faster than a slow-slewing op-amp, and doesn't rely on a precise voltage applied across the 10mΩ current sense resistance.
Don't forget to account for the \$R_{DS(ON)}\$ of the MOSFET, which will contribute to the total resistance in that path. You'll find you may need to raise the supply a little to compensate for a non-zero \$V_{DS}\$.
If you must use 2V, then consider limiting current with another resistor, to obtain the expected 5A:
simulate this circuit
The same caveat regarding \$R_{DS(ON)}\$ still applies, but instead of reducing the supply voltage, you could reduce R2 instead.