Makes sense. After all, the motto of Italian fascism was, "We move Germans." I'm sure they'd be happy to convert their industrial production over to churning out support ships for a foreign power.
While you're at it, maybe if they ask nicely the Japanese will send their fleet to the party too.
Ignoring the fact that Italy was the subordinate power even IOTL I see.
If Italy has any sense, they'll realize they can't take the RN alone. Hell, only the United States and Japan might have been able to do it on their own.
And they'll be fighting the UK if the Axis gets sucked in.
So Italy can either get her ass kicked around the Mediterranean, or back the Germans if they think they have a shot.
Frankly German-Italian cooperation should have been much greater from the start. Germany tools up Italy to build panzers, free exchange of weapon military technology, joint war games, etc.
Italy needed the leg up, Germany could have benefited from their experience with aviation in the mid 30's.
So, yeah, Italy would absolutely build landing craft if you approach it correctly. "Hey Benny, sure would be handy if you could drop troops right on the Greek coast and avoid all those mountains. Maybe we could use a few transports and landing craft".
As far as the naval cooperation goes, if Germany bags the French fleet, and Hitler tells him they're going to try and cripple the Royal Navy, and that they wanw the Italians to be a part of it, Mussolini would jump on that in a heart beat.