The answer, by the way, is no. Facebook has had highly-accurate "shadow profiles" for people who do not use their service for over a decade.
Having a basic interaction graph, geo-location data, and being in the income band able to afford a $1500 headset begins to narrow down people a lot faster than you think. Not to mention the headset is technically capable of gathering skin tone (easily mapped by the hand recognition), IP address, height, and even performing facial recognition.
Using a fake username achieves nothing. Unless you use it for sideloaded, offline, single player games and nothing else - at which point, the pro headset doesn't really make sense.
What about phone number and IP?
Your friends list also probably uniquely links the account to your shadow profile which has your real name and address bought from data brokers.
Don’t forget that Facebook has your name/phone number from all of your friends’ phones who gave FB/WhatsApp contacts permission. Friend enough of them and they’ll know it’s you.
Also, they have the plaintext of all of your chats with your friends, including that initial “hey it’s me bob” intro message.