Using the existing state of art of the QCD expressions of the two-point correlators into the Inverse Laplace sum rules (LSR) within stability criteria, we present a first analysis of the spectra and decay constants of $B_c$-like scalar $\left(0^{++}\right)$ and axial-vector $\left(1^{++}\right)$ mesons and revisit the ones of the $B_c^{⁎}\left(1^{--}\right)$ vector meson. Improved predictions are obtained by combining these LSR results with some mass-splittings from Heavy Quark Symmetry (HQS). We complete the analysis by revisiting the $B_0^{⁎}\left(0^{++}\right)$ mass which might be likely identified with the $B_J^{⁎}\left(5732\right)$ experimental candidate. The results for the spectra collected in Table 2 are compared with some recent lattice and potential models ones. New estimates of the decay constants are given in Table 3.