Yes that should work.
Though what I generally do is take, in the cooler, a cooling jacket designed for a wine bottle, which I keep in my freezer. That works surprisingly quickly, but only for wine bottles, as the dimensions are designed for one to fit inside snugly, making a good thermal contact. And I only have one, so only good for a single bottle.
Freezing point depression occurs because the dissolved salt ions add entropy to the system, which makes the entropy change on freezing even more -ve than it is for pure water, thereby making it more of an "uphill"process and hence requiring a lower temperature to bring it about.
In terms of the good old chemist's thermodynamic formula, ΔG = ΔH -TΔS, ΔG must be -ve for the process to occur, i.e. the free energy of the system after the change must be less than before, as it moves to a lower energy state.
ΔH is the Latent Heat of Fusion, which is -ve for freezing because heat comes out, i.e. less is left afterwards. So that term favours freezing.
However ΔS is also -ve, because the molecules have to arrange themselves into a highly ordered state in the crystal, so the crystals have a lower entropy than the liquid. Since minus of a minus is a plus, the -TΔS part is +ve and fights against the ΔH term. Which one wins depends on T, the (absolute) temperature.
Since -TΔS is more +ve for salty water (not only do the water molecules now have to order themselves to make a crystal, they also have to get all the salt ions out of the way to do so), T has to be lower to make the -TΔS term small enough for ΔH to win.