The HMI also handles switching of the rear camera input/ output, so the fact that the backup camera works on demand generally rules out the HMI as well. If the HMI fails, you'll still have sound coming from the radio, but the infotainment will be blank or unresponsive. Typically, stations and volume can be controlled from the instrument cluster if that happens.
The radio, amp, HMI, CD player, and instrument cluster are on their own dedicated communication bus known as the Media Oriented Systems Transport, or MOST bus. It is fairly easy to diagnose with the proper tooling. GDS2 and some aftermarket scan tools will outline which 'node' or module (of the aforementioned components) are not working properly and help to isolate the defective module or circuit portion. The dealer should have been able to find this out without too much effort.