Statistical models involving bivariate data are among the most important areas of statistical theory. Concomitants of $k$-record values (CKR) are key topics in statistical theory that have not been extensively studied by researchers. In this paper, we derived the marginal distribution of CKR based on the Huang-Kotz Farlie-Gumbel-Morgenstern type Ⅱ (HK–FGM2) family of bivariate distributions. Additionally, we obtained an expression for past extropy associated with the CKR within the HK–FGM2 family. Three related uncertainty measures (the cumulative residual extropy (CREX), cumulative extropy (CEX), and the negative cumulative extropy (NCEX)), were formulated for CKR based on this family. Furthermore, estimation techniques for CREX and NCEX were explored by employing empirical estimators tailored to CKR using the HK–FGM2 family. Finally, we analyzed a computer series system dataset for illustration purposes, providing significant insights into the behavior of CKR uncertainty measures.