After dealing with the Dragonborn Cultist encounter, I traveled around with the helmet in my inventory for quite a while. Then I tried outfitting the upper floor Lakeside Manor Armor Mannequin with full Cultist gear. Only the Robes went on the mannequin. The boots and gloves went into its inventory, but the helmet would just vanish and reappear instantly back in my inventory.

After that I tried putting it in various containers, both in Lakeside and other locations. It appears back in my inventory after a while. I haven't checked it if vanishes from the containers, but I'm pretty sure it doesn't.

I also threw it on the ground in two different places, and just now when I went to sell some stuff it was back in my inventory. How do I get rid of it?

Skyrim with all DLCs as well as a handful of mods - SkyRe with Immersive Armors, Weapons and Armonizer, plus the Unofficial patches.