Fermentation Chamber question

Alright, worried a little....

I am getting ready to brew my first AG, an Oktoberfest, and I have ran into a little problem. I have a fridge in the garage and I have a temp controller for the fridge, so the fermentation is no problem.

My problem arises because we are experiencing unseasonably warm weather (95-100) right now, even before summer, plus I will have to Lager through Summer heat. Where this plays havoc is my fridge is set up in the garage because SWMBO will not let me bring it into the house, which is fine, except the fridge is just an apartment sized fridge so the compressor has a lot of trouble keeping up with the 85-90 degrees inside the garage throughout the day. So the coldest I can get the fridge is about 45-50, perfect for the fermentation of the lager, but crappy for the lagering.

Now the question: I have seen many peoples threads about building fermentation chambers out of old fridges, mainly what I would call a dorm fridge. I moved into a new house about a year ago and the previous owner left me a fridge similar to this, a small dorm refridgerator with an inside freezer.

My question is I can build this into a fermentation chamber with no problem, and also I have an empty storage unit in the basement that I can place this so the compressor will not work nearly as hard, but how cold would I be able to get the fermentation chamber with a similar fridge like this?

The fridge is currently running right now and I can get temps extremely close to freezing, I'm just wondering if I could expect this out of a fermentation chamber if I insulate it correctly.