tim2akat2 is right on both counts.
Having followed a few build threads using similar cold-plate units your options are pretty much limited to unfastening the cold-plate from the "roof" and carefully bending it flat to the back wall.
The thermostat sensor probably won't reach so you'll likely have to remove it from the plate and find some place to hang it. That will affect the internal controller's ability to regulate (apparently it really wants the direct sensing of the cold-plate temperature, reading the air temperature isn't enough) but that's not a huge deal if you're going to use an add-on controller.
You might be able to Google up a thread or two to get some ideas if you search the model number of your fridge...