I really like the hook, which can easily be delivered by any patron, although I imagine it coming from the basilisk's mouth. Like the hook, I bet ripping the heart from her chest would a satisfying conclusion.
The cannibalism and ghouls in a snowed in town though feels trope heavy. I think that fleshing out some interesting repurcussions to the town after being beset by winter for so long would be valuable and make it feel more unique. Or maybe a why about the Mayor's daughter having this burning heart?
Maybe the reason it's so cold and wintery is because she has somehow drawn the warmth out the landscape into herself or something. There's a lot of potential in the Burning Heart.