You can fly GOA from LGW on BA as well. Having Genoa as a base for the trip you can go to Langhe with a 2 hours drive (not a great impact in winter, but food and wine are always good). You can go to Cinque Terre (but I would suggest train in this case, or you go to Levanto by car, park it there and them use train between the Terrre). Portofino is off limits as for now. Again, you can take a car and drive to Pisa, Lucca or Florence. You can drive to Sanremo and this coast is awesome.
Pros: it may snow here in Genoa, but it happens once in a winter time. The climate is mild and it is not so cold as in other parts of Italy. It is really a 2 hours drive to other sightseeing places.
Contros: do not consider bridge collapse as a contro. If you drive away in early moring you won't sufer heavy traffic.
Consider that Appenine area (Bologna, Florence) can we affected by heavy snow in winter.