The DB Initiate followers love wearing their cowls and seem to refuse to wear a circlet. Here's what I did to finally make mine drop her cowl for good.

- Initiate is wearing something other (better) than the cowl. This puts the cowl in inventory.
- Initiate is following you.
- Dismiss the Initiate.
- Pickpocket the cowl.
- Command the Initiate to follow you again.
- Access inventory, take their current headgear and try giving the circlet to the Initiate.
- Initiate reverts back to cowl.
- Give the non-cowl headgear back, cowl is unequipped.
- Dismiss the Initiate.
- Pickpocket the second cowl that suddenly appeared in the Inventory.
- Command the Initiate to follow you again.
- Access inventory, take their current headgear again, then try giving the circlet to the Initiate.
- Circlet will now be equipped (or no headgear at all, if you wish.)

That's how it worked for me, I only had to steal two DB cowls. It was bothering me, so I had to figure this thing out.