As a godlike, can't equip the hood from your inventory. When you enter the room from the 'Catacombs' entrance a dialogue pops up asking you whether you want to don the hood and enter the room alone, don the hood and enter it with your party, or not don the hood and enter it alone or with your party respectively.
This dialogue and the corresponding options also obtains for godlike, which normally cannot don any kind of hood or helmet.
(edit: I tried this with my male moon godlike and entered the room with donned hood alone, thus preventing conflict)
Thanks--responded to another similar reply . . . finally got this result. The option didn't pop up in the dialogue the first few times I tried . . . then, it did, and I was in. Felt like a dunce--in need of a dunce cap--but, hey, live and learn . . . and living is good.