The living shadow is not a creature. It doesn't have hit points or an armor class, and most importantly, it always persists and there are no end conditions listed in its description:
The shadow you cast is animate and ever-present, even when lighting conditions would otherwise prevent it.
There isn't much else to say here, there is just nothing in the gift description that gives any indication that firing a crossbow bolt at it, enchanted or otherwise, will do anything.
The shadow probably isn't an eligible target for an attack.
The rules for making an attack require that we first select a target:
Pick a target within your attack's range: a creature, an object, or a location.
The rules give no indication that the shadow is a creature or an object, despite the shadow exhibiting some behaviors we might see from a creature. So at best, we could declare our target to be something like "the shadow's current location on the ground", but as mentioned previously, the rules for the living shadow give no indication that this would have any effect.