Wendy, yes its a very good idea to build on the Italian trees you have started, however small and wide, because they will eventually act as a fishing net and will bring in more, wider matches to them. Eventually they could attract matches in the direction you need, closer to your grandfather's immediate family, which will then provide a solid surname. Without useful online Italian resources, the DNA numbers will help too, so keep your eye on the closeness scale numbers. Can't remember their technical name.
Meanwhile the connections you have to Bardi, which is in the Parma province, is exactly the kind of place where local knowledge will be abundant. Businesses like cafes, bakeries or ice cream parlours etc are often held and passed down through generations of the same family in Italy. So always start with asking those if you end up going once you have a solid surname for your grandfather. We passed by the top of Parma province, in Parma itself, on our foodie tour of Italy. Very easy area to travel through by car, lots of little roadside cafes have rooms to let above, no booking required. We just pointed the car and drove, and was never short of a place to stay, even at 9 pm at night. A lot easier than in France, and more affable natives, who ensure you go to bed full of Limoncello on the house, once they find out you are of Italian descent looking for your relatives!