IMO, its always good to go with soil for your first grow, especially in a new cabinet. You're introducing too many variables with hydro, particularly if you don't have a lot of experience with it.
Edit: Soil is just a lot more forgiving medium to grow in.
Also, even aside from the temperature issue, I'm concerned about ventilation.
Yes, you have two fans going, which ought to be enough to ventilate the small box, but the problem is that you're ventilating the smaller box into the larger cabinet. In other words, you're basically just pushing hot, stale, humid air around the inside of the kitchen cabinet. How are your plants going to get CO2 and O2 if they're effectively growing in a closed box?
You need to make sure cool fresh air can get inside the kitchen cab, so it can get into your small growing box.
Incidentally, if you're already growing stealthily inside the cabinet, why not convert THAT into your grow space? Why a box-within-a-box? You'll have a lot more room to work with, and it will also probably be more forgiving if you go with the bigger cab as your grow space.