I've only played with a Peltier module a few years ago but I recall one issue is the difference between cooling and losses.
Cooling is a function of average current. Losses are a function of the RMS current.
So if you have 12V and you need the cooling a 6V supply would generate....
Cooling approx = to that with 6V.
Losses will be Ploss = V^2/R so during the 12V period the losses are 4 times that of 6v. And because a 50% dutycycle the losses are down to 2 times that of a 6 volt supply.
Because Peltier's are so inefficient I would think extra losses are not desirable.
Note there may be other reasons not to PWM a Peltier I don't know. I'm not a good source for Peltier operation except for the above.