That chip is rated at 1A, so I think I'd need a 10 ohm 10 W resistor in series to limit the current.
A 10R resistor across 12V gives 1.2A. Yes I know some voltage is dropped across the Peltier but don't you think you are running the H bridge too close to it's maximum rating, it's best practice to use a device at only 80% of maximum rated value.
Most devices like this I have seen use forced air cooling on the outside of the heat sink.
When the temperature difference across a Peltier exceeds a certain value it goes into heating mode rather than cooling so you need to shut down at this point.
Otherwise is sounds good.