You were right!
I got a set of Hide Armor and wore the Fur Helmet (and vice-versa) and the set bonus worked. So I added ArmorMaterialHide to my Fur Circlet and now it grants the exact same armor bonuses that Fur Helmet does (to both Fur Armor and Hide Armor!) As far as I know, my Fur Circlet is now behaving exactly like a Fur Helmet, except not looking awful.
Big thanks to GreyFox77, SnippyChris, and Lofgren for your help!
PS: As for Complete Crafting Overhaul Remade, it looks great, but I already use Circlet Helms and Dragon Priest Circlets, Lost Art of the Blacksmith, and Legendary Smithing Upgrades. I've been using these for hundreds of real time hours in the game, and I have also crafted circlets, enchanted to my preference, and upgraded to legendary before resetting my skills for enchanting and smithing, so I can't currently replicate the circlets for my sets in the new mod until I retrain and reperk my skills... actually, I just remembered I simultaneously enchanted and improved the helmets at the same time just in case the circlets were unavailable for some reason while I was retraining. I'm a genius! I'd assume the smithing mods would conflict with this one, but I also see that A.) They're years out of date, and B.) They don't have nearly the same features. I want to switch over to this one, but before I do, do you have any recommendations to do it safely?