Yours might be a case in which it would be worthwhile taking ketone supplements.
But don’t worry about your ketone levels, as long as you are keeping your carb intake low. Circulating β-hydroxybutyrate levels represent the gap between production and consumption; there is no way to measure production directly. The main benefit of having measurable ketones is the proof that our insulin is low enough for our metabolism to be relying on fatty acids, rather than glucose.
As far as oils go, you might want to avoid oils in favour of animal fats: butter, lard, tallow, and bacon grease. The reason is that they are mostly saturated and mono-unsaturated fats, and what polyunsaturated fats are ones that our systems have evolved to use. The high polyunsaturated fat content of the industrial seed oils is made up of fatty acids that are pretty much unknown in our evolutionary history, and they have unpredictable effects in the body, such as when they are taken into our cell membranes, for instance.
Moreover, the phytosterol content of the seed oils is another problem in the the body. Our body can easily make cholesterol and ketones from animal fats, but has trouble with the polyunsaturates from seed oils. Fortunately, the brain makes its own cholesterol from the ketones that make it across the blood-brain barrier (fatty acids are too large to cross), which is another reason to eat fats that can feed the liver’s ketone-making machinery. (Ketones are intermediate products in the process of fatty-acid metabolism.)