PWM can work but it's usually a bad idea...
The Peltier would be fine but a "regular" 2 Amp power supply will burn-up if you connect a 5A load. 
If you run it at a constant percentage the heating (or cooling) will be constant but not the temperature... If the temperature in the room goes up by 2 degrees the Peltier will also go up by 2 degrees.
Heat can't change instantly and almost every heating or cooling system works by power cycling and feedback. If the temperature in your home gets too low, the heat comes on until it reaches the target temperature, then it turns off. Your oven, your water heater, your refrigerator, air conditioning, ALL work like this.
There is usually some hysteresis (or "swing") so it doesn't turn on & off several times in a second.
If you try to use PWM (or any percentage control) it can be very tricky and the unavoidable delay can cause instability where it's constantly overshooting and undershooting and "hunting" for the right temperature and/or it can end-up responding too slowly.