A 52-year-old man with a family history of premature coronary artery disease presents with xanthomas on his Achilles tendons and eyelids. Fasting lipid panel shows total cholesterol 480 mg/dL, LDL-C 420 mg/dL, and triglycerides 150 mg/dL. Clinical suspicion for familial hypercholesterolaemia is high. Which investigation would be most appropriate to confirm the diagnosis and assess the underlying genetic defect?