I started keto about a week ago and immediately had an attack. That sent me digging and questioning my priors, and deciding it wasn't for me.
I started with a 36 hour fast and I was definitely in keto without the "flu" people complain about. Then I ate keto only the next day, unfortunately by eating bacon and fatty lamb which set off the attack.
Technically keto is from a lack of carbs. A water fast puts you into keto. So it's not strictly incompatible with gout. But to practice it long-term you need to replace the carbs with something, and usually that is fat. If it's a bottle of canola oil you won't get gout, but if it is fatty meat you might. But there are other reasons not to go drinking canola oil.