I did Car Audio install professionally for 4 years. I still think the issue is that the Limited headunit doesn't have a built in amplifier. It probably has some very low power connector that goes to an external amp, like this one http://www.ebay.com/itm/like/171865901065?ul_noapp=true&chn=ps&lpid=82 which then powers the speakers. Whereas the non-Limited Headunit most likely has a small internal amplifier to power the speakers in the Edge.
If you replace the Headunit with the Limited version and buy an external amp like the one I linked to you could possibly get audio. It depends on if the harness that goes from the Limited headunit to the external amp is in the car, but just taped off somewhere unused at the factory. Not only would you have to connect the Limited headunit to the external amp, you would have to connect your speakers to the external amp. This could be as simple as taking the wiring harness for the speakers that plugs into the back of the non-Limited headunit and, if the connector is the same, plugging it into the output of the external amp. If the connector is different, you'll have to find the connector somewhere online, and splice it together with the wiring harness that currently goes to the speakers. Note that there will be extra wires as the Limited audio system has a center channel and subwoofer, perhaps more speakers than that, not sure.
This is just my theory, I haven't verified any of this, but in my experience this is how every other "Premium" audio system works in other vehicles.