Familial Hypercholesterolemia (FH) is a genetic disorder characterized by elevated levels of Low-Density Lipoprotein (LDL) cholesterol or its associated genes. Early-stage and accurate categorization of FH is of significance allowing for timely interventions to mitigate the risk of life-threatening conditions. Conventional diagnosis approach, however, is complex, costly, and a challenging interpretation task even for experienced clinicians resulting in high underdiagnosis rates. Although there has been a recent surge of interest in using Machine Learning (ML) models for early FH detection, existing solutions only consider a binary classification task solely using classical ML models. Despite its significance, application of Deep Learning (DL) for FH detection is in its infancy, possibly, due to categorical nature of the underlying clinical data. The paper addresses this gap by introducing the FH-TabNet, which is a multi-stage tabular DL network for multi-class (Definite, Probable, Possible, and Unlikely) FH detection. The FH-TabNet initially involves applying a deep tabular data learning architecture (TabNet) for primary categorization into healthy (Possible/Unlikely) and patient (Probable/Definite) classes. Subsequently, independent TabNet classifiers are applied to each subgroup, enabling refined classification. The model's performance is evaluated through 5-fold cross-validation illustrating superior performance in categorizing FH patients, particularly in the challenging low-prevalence subcategories.