  We present a comprehensive analysis of the semileptonic weak decays of $B_c$
meson decaying to axial-vector ($A$) mesons for bottom-conserving and
bottom-changing decay modes. We employ self-consistent covariant light-front
quark model (CLFQM) that uses type-II correspondence to eliminate
inconsistencies in the traditional type-I CLFQM. As a fresh attempt, we test
the self-consistency in CLFQM through type-II correspondence for $B_c \to A$
meson transition form factors. We establish that in type-II correspondence the
form factors for longitudinal and transverse polarization states are
numerically equal and are free from zero-mode contributions, which confirms the
self-consistency of type-II correspondence for $B_c \to A$ transition form
factors. Furthermore, we ascertain that the problems of inconsistency and
violation of covariance of CLFQM within the type-I correspondence are resolved
in type-II correspondence for $B_c \to A$ transitions. We thoroughly
investigate the effects of self-consistency between type-I and type-II schemes
using a comparative analysis. We also study the $q^2$ dependence of the form
factors in weak hadronic currents for the whole accessible kinematic range $0
\leqslant q^2 \leqslant q^2_{max}$ for both bottom-conserving as well as
bottom-changing transitions. In addition, we extend our analysis to predict the
branching ratios of the semileptonic weak decays of $B_c$ meson involving
axial-vector meson in the final state to quantify the effects of
self-consistency in these decays that were not studied before. We evaluate the
lepton mass effect on these branching ratios and various other important
physical observables, such as forward-backward asymmetries, lepton-side
convexity parameter, asymmetry parameter, and longitudinal polarization
asymmetries and fractions. Finally, we obtain the lepton flavor universality
ratios for various decays.
