One solution I found was to move LPopup into the parent file. This worked...but I have no clue why this worked and the aforementioned code doesn't.

Workaround:

Parent:

<LMarker
    v-for="listing in listings"
        :latlng="[listing.lat, listing.lon]"
>
    <LPopup>
      <ChildVue />
    </LPopup> 
</LMarker>

Child:

<template>
   <p>Hello world!</p>
</template>