If I shaved a white chocolate bar and put the shavings in a cake batter, would they melt or harden?
From my experience it's likely that chocolate shavings are too small to keep any integrity during baking. They will likely melt into the batter as it bakes. If the batter is warm it may already melt then.
Larger pieces like chocolate chips are more likely to stay together enough to re-solidify into noticeable pieces when eaten.
Chocolate (white or not) cannot "harden" much from baking, unless it is essentially burned. Things that "harden" when cooked/baked do so by either drying (chocolate is already dry) or by proteins/starches/sugars changing (chocolate has not much protein or starch, and at the temperatures where sugars change chemically, chocolate is likely to burn). The more unfortunate possibilities would be fats polymerizing (you don't want that happening to your chocolate either) or pyrolizing (hardening because ash is hard).
However, chocolate that has been molten and resolified will usually be more matte and softer than before it (since it is no longer properly tempered).