Hmm ... I don't know from personal experience, but let me try to offer an idea or two anyway. If I recall correctly, the pH of the cheese has an effect on how well the cheese will melt. Unfortunately, I don't recall exactly what the relationship is, e.g., if the pH goes too low.
When you say "blitzing," do you mean something like using a blender or food processor? I would expect that this would give you a grainy result, but it would probably taste fine. My first thought, though, would be to experiment with something like a fondue approach. As noted above, it might not work, depending on the pH, but I'd try with just a little of the cheese. Heat up some white wine or clear broth, and grate some of the cheese in to melt. If it doesn't want to incorporate smoothly, try tossing the grated cheese with a little bit of flour or cornstarch before adding to the wine or broth. If it still doesn't want to incorporate ... time for plan B.
