  The measurement of the $^7$Be($d$,$^3$He)$^6$Li* transfer cross section at 5
MeV/u is carried out. The population of the 2.186 MeV excited state of $^6$Li
in this reaction channel is observed for the first time. The experimental
angular distributions have been analyzed in the finite range DWBA and
coupled-channel frameworks. The effect of the $^7$Be($d$,$^3$He)$^6$Li reaction
on both the $^6$Li and $^7$Li abundances are investigated at the relevant
big-bang nucleosynthesis energies. The excitation function is calculated by
TALYS and normalized to the experimental data. The $S$ factor of the
($d$,$^3$He) channel from the present work is about 50$\%$ lower than existing
data at nearby energies. At big-bang energies, the $S$ factor is about three
orders of magnitude smaller than that of the ($d,p$) channel. The ($d$,$^3$He)
reaction rate is found to have a less than 0.1$\%$ effect on the $^{6,7}$Li
abundances.
