Not gamed that one out, or done much research on it, but would like to. Gaming out ANVIL type operations of April 1944, like Eisenhower wanted, or a earlier January 1944 execution made things 'difficult' for the German side.
Politically its difficult since first the Brits became focused on taking Sicily to reopen the Mediterranean sea route, & then Churchill was infatuated with securing Italy and great things in the Balkans, the Lubjana gap and all that. But, assuming the Allies do decide to forego Op HUSKY & all that followed in that direction:
A. Sardinia/Corsica were poorly defended in the spring and summer of 1943. With air support from Algeria its possible to overrun those islands even as Tunisia is in the final stages of battle. OTL the Joint Chiefs proposed Sardinia be invaded as early as March 1943. If its made a priority its possible as early as May or even April 1943. OTL it took the US Air Force support services two months to establish a base structure for 1000 aircraft on Corsica, in January 1944. If the resources put into developing the south Italy airfields are directed to Sardinia/Corsica its not unreasonable to think 2000 to 3000 aircraft can be based there mid 1943. That puts a lot of southern Europe in range of Allied medium bombers and fighter cover.
B. Sans the Italian campaign a rather robust Allied army group is at large & unemployed. In July 1943 Op HUSKY placed that AG ashore in the second largest amphib op of the MTO/ETO. It does look like there are possibilities in that context for S France.
One payoff for the Allies is the need to feed and support the liberated Italian population is waived away. 1943-44 feeding and providing coal to the impoverished provinces of south Italy sucked away a uncomfortable chunk of Allied cargo shipping. I don't know if that would be the case where Frances southern Maritime provinces are liberated.
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What would happen to Italy?
Italy continues a economic death spiral, the war is increasingly unpopular, the Facist Grand Council finally removes Mussolini, the surrender option is eventually choosen, the Germans support a counter coup and try to maintain a Facist Italian state. At some point the Allies are sucked into some sort of secondary Italian campaign with a much smaller force than OTL. Postwar the details are a bit different for Italy, but the major trends are the same.