Was it
required to wear the Chef's Tunic and Chef's Hat? I just put it on to be funny so I got in on my 1st try - they turn you away if you're
not wearing it?

I think the Writ of Passage will get you past the Penitus Oculatus regardless, but I could've sworn I read somewhere that you needed to be wearing chef's clothes to interact properly with the castle cook who actually makes the dish. I would've worn them anyway because: lulz. And also because I find it ridiculous that I can waltz around sometimes in clothes/armor that should ID me instantly as The Enemy of certain people, and no one who should care really does... they might comment on it but it never really impedes my progress much.
So if I have to go undercover, I make myself dress appropriately, for instance, I didn't wear my Stormcloak armor when trying to infiltrate Understone keep and sneak around the steward's chamber in order to blackmail him over his Talos worship. No, I did it in a full suit of finely crafted (by me) Imperial heavy armor, complete with an officer's helmet.
Let the old fool think he was blackmailed by an officer of the very Legion his Jarl is supporting, that just adds to the fun.
