First, in OTL the two Mulberry harbors consisted of 400 components (including the port infrastructure and ship and cassions for the breakwater). Even if you're only doing one Mulberry that's still 200 old hulks and large lumps of concrete you need to tow... for a once off effort ala D-Day you'll be needing at least 50 ships put aside to do the towing.
Second, where do you prefabricate the components? There's little point doing so in North Africa since now you've just tied up the existing port infrastructure bringing the concrete and steel in. If you do so in Italy itself, well, Catania to Tripoli is around 300 nautical miles, to Benghazi 450 nautical miles. Using a major port on the Italian Peninsular itself puts distances around 500-600 nautical miles. By way of comparison during Overlord the Mulberrys had to cross somewhere between 100 and 200 nautical miles, and even then it took something like two days to get the first components into position and around a week to bring the ports to operational status. The greater distance means the Italians either need to put more vessels on towing duty (can they spare the necessary numbers?) or they need to accept a longer construction time since the vessels involved need to cover a larger distance, this in trun brings an increased risk of detection and attack.
Third, given prefrabrication in Italy the Italians need to tow through open waters, unlike the allies during Overlord. This means they're more exposed to both weather and surface attack.
All in all, it looks to me like you'd need naval supremacy to pull it off.