Controlling heater with PWM through MOSFET
It's probably not mostly from the inductance. 

More likely, pulling close to 8 Amps from the battery has a significant effect on the battery voltage, and this changes the switching thresholds around the comparator generating the PWM signal.

You probably need to feed the LM393 and R3 from a lower noise supply, either R-C filtered (say 50 ohms and 1000 uf) from the battery, or perhaps better, from a 5V LDO regulator (with decoupling). 

You can keep the pullup resistor R1 connected to the full battery voltage to turn on the FET as hard as possible, even with the LM393 supplied from 5V.

And as the voltage peaks exceed the battery voltage, inductance must be having some effect so the flyback diode is definitely recommended.