Only avoid red meat if you have a personal aversion to it. And presuming you are thinking you should avoid it because it is bad for you… watch this video and I highly recommend reading her book (or listening to the audiobook), “The Big Fat Surprise”. It is FANTASTIC and a great eye opener.
Also, keep in mind that you can eat chicken all you want and still get into ketosis. What gets you into ketosis isn’t the amount of fat fat you are eating, it is the number of carbs you aren’t. A couple of months ago I went to an AYCE Asian Buffet. What seriously lowered my ketosis levels for the next few days wasn’t the lack of fat - that one meal was 283g by itself - but rather the 622g of carbs that came with it.
The average American get’s plenty of fat, but what keeps them out of ketosis is the number & frequency of the carbohydrates they are eating.
What the fat does is help get you satiated and not need to eat. Keeping the carbs low keeps your insulin down. When your insulin is low for extended periods of time your body can access your fat stores and break down the fat to produce the ketones. The longer you keep your body in this state the more your body learns to use it better. Eating the fat simply helps satiate you to keep you from feeling hungry longer and gives your body more of the fuel source you are trying to get it to use all the time.