There was one game where I had a damaged Italian BB at S. France and it was German owned. The UK player sbr’ed the harbour, preventing me from repairing my BB and using the harbour to move from there to Egypt SZ. So that might another downside to taking it with Germany, along with the weakened attack on France and the fact that the French med fleet can convoy damage your PUs, unlike with Normandy where you’re guaranteed the 2 PUs.
Generally speaking though, I also don’t like diverting Italian ground forces to take S. France, so I almost always launch an attack against it on G1. Once I retreated after killing one unit and then moved in and inf and art to take it with Italy. But in general, I take it with Germany and keep it. Having the option to build german boats in the med is always good.