Caramel made with coconut milk tastes burnt
Try coconut cream.

I have moved away from coconut milk except for soups.  Coconut cream is also in cans and is lovely, with much less wateriness and great flavor.  It is often semisolid right out of the can but melts in the pan.  The mammal miracle is that milk holds fat in suspension.  I suspect this is trickier for coconuts.  The cream is fattier and so whatever they use for an emulsifier is present in greater amounts.

Try your veggy caramel with coconut cream and report back!