Light-cone sum rules with $B$-meson distribution amplitudes for the $B\to p$ form factors in $B$-mesogenesis models

New decay modes of $B$-meson into a baryon and invisible dark antibaryon $\Psi$ are among the most distinctive signatures of the $B$-mesogenesis scenario. We concentrate on the proton mode and consider two versions of the underlying interaction of $\Psi$ with quarks, the so-called models $(d)$ and $(b)$. To estimate the width of the $B^+\to p \Psi$ decay, we obtain the $B^+\to p$ transition form factors, applying QCD light-cone sum rules (LCSRs) with $B$-meson distribution amplitudes with an accuracy up to twist-5, while interpolating the proton with a current. This method is independent of the previously applied one, which was based on the nucleon distribution amplitudes. We estimate the partial width of the $B^+\to p \Psi$ decay as a function of the dark antibaryon mass. Furthermore, we use the ratio of this width to the inclusive $B\to X_N \Psi$ width, the latter predicted earlier using the heavy quark expansion method. This ratio, which is independent of the effective coupling, when combined with the minimal inclusive branching fraction of $O(10^{-4})$, necessary for the feasibility of $B$-mesogenesis, yields lower limits on the $B^+\to p \Psi$ branching fraction. We confront these limits with the most recent upper bounds obtained from BaBar and Belle/Belle II searches for the decays of $B^+$-meson into a proton and missing energy. The comparison indicates that experimental upper bounds on the branching fraction of $B\to p \Psi$ at the level of $10^{-8}-10^{-7}$ are needed for a decisive probe of this invisible mode of $B$ decays.