Unofficial Skyrim Patch: Wearing a hooded robe and a dragon priest mask at the same time

When I do not have the Unofficial Skyrim Patch installed, I can wear the Archmage's Robes and Morokei at the same time (which is not a bug/glitch, it's intentional).

After installing the Unofficial Skyrim Patch, this combination is no longer possible. If I equip the one and then the other, the first gets unequipped like they are not compatible.

I can only guess the authors of the mod introduced this change as a reaction to this Skyrim bug:

[PC] [360] While wearing the robes, Dragon Priest masks (such as Morokei and the Wooden Mask) might make the player's head invisible, to solve this, drop the robes and pick them up, or unequip the mask.

How can I make this combination possible without having to give up on the fixes introduced by the Unofficial Skyrim Patch?

Based on information from UESP on Archemage's Robes, the version of the robes you got are "hooded" and isn't suppose to be worn with a headgear (see Bugs section in UESP).

However, there is a second version that is non-hooded and is available in the Archmage's quarters. The Unofficial patch corrects this version and allows you to wear it along with Morokei.