MOSFET getting hotter than calculated
You are assuming that the current is constant. In fact if your supply has ripple, the dissipation will be much higher than if the voltage is steady, because the power is the integral over time of the currents squared times the Rds(on) resistance (which might be 45m\$\Omega\$ as others have pointed out). If you have a true-RMS meter you can measure the RMS current and voltage (current is more important than voltage across the device, but voltage through a sense resistor is fine) and determine the heating effect. 

It will also change with the temperature of the Peltier so you may not have done an exact A-B comparison. 

Even at 45m ohms, you should probably have a small heat sink on the MOSFET or get a better MOSFET, if the current is in the 4A RMS region.  

By the way, ripple which causes the MOSFET to heat will also seriously negatively impact the efficiency (miserable at the best of times) of the Peltier device. It should certainly be kept to less than 10%.