It depends on several factors, for example, I'd stay away from
homoenized or powdered dairy, or dairy with carrageenan in it.  Organic
eggs are okay if boiled.  It's all about oxidative stress.  If the food
enhances this stress, it's a big potential problem, if not, it's okay,
unless it contains large amounts of something that is a potential
problem in large amounts but not in small amounts.  For example, turkey
is high in tryptophan.  Small amounts of problematic foods cooked
correctly, while eating foods that are generally okay in large amounts,
such as organic blueberries, unrefined coconut oil, etc. is the way to
go.  I don't have any interest in the low carb diet myself, though,
because obesity is usually an inflammatory process gone chronic, and
carbs or no carbs, it's not a healthy situation.  Of course, cutting
calories dramatically will most likely drop pounds, but you could be
doing so in a way that creates massive free radical activity in your
body, which could lead to neurodegenerative "diseases."  Smart thing to
do is to stick with the safe foods and drop the weight gradually as the
inflammation recedes - it won't happen overnight, but within a few
months you should see a big difference.

You need to read your Atkins books before embarking on your imagined diet.

You cannot eat cheese all day, There are limits and you need to eat 1.5 cups
of vegetables. It is not carb free.