One obvious question to ask is, what subtypes of fat (saturation, chain length, etc) are in the triglycerides the adipose tissues are storing? That is, take an adipose tissue biopsy, separate out the triglycerides then separate those into free fatty acids, and measure fractions of saturated, monounsaturated, omega-3/6/9, etc. fats, so that if one subtype has abnormally high concentration you can look at why it might not be metabolized properly. Unfortunately, this does not appear to be a standard test, so it might be hard to find someone to do it.