It turns out I’m not the first person to misidentify a bee louse. Bee lice are imports from Europe and they live in honey bee colonies. They are insects and so have six legs, unlike the Varroa mite which is an arachnid possessing eight legs. The bee louse, Braula coeca, is actually a small wingless reddish brown fly less than 1.5 mm in length. They are considered to have only a minor impact on colonies, so little attention is paid to them.
The adults are commonly found on the heads of honey bees where they seek bits of food as it is transferred between bees. Generally, workers carry only one mite, but drones may have more, and queens may have many. According to the University of Florida, a single queen has been found to carry 180 lice at one time.
The adult female may lay her eggs in many different places within a hive, but in order to hatch, the eggs must be placed on top of honey cappings. After hatching, the larvae tunnel under the cappings, eat wax and pollen, and leave a visible trail.
It turns out that many Varroa control products also kill bee lice, so they are most often found in hives that have not been treated for Varroa.
In the photos below from the University of Florida website, you can see a comparison of an adult bee louse on the left and an adult Varroa mite on the right.