mg.metric geometry - Bounding the product of lengths of basis vectors of a unimodular lattice

I don't know how good the bound is you can obtain from this, but what about taking a Korkine-Zolotarev reduced basis of $Lambda$, say $(b_1, dots, b_n)$: then, by this paper, $|b_i|_2^2 le frac{i + 3}{4} lambda_i(Lambda)^2$, where $lambda_i(Lambda)$ is the $i$-th successive minimum of $Lambda$. By Minkowski, $prod_{i=1}^n lambda_i(Lambda) le gamma_n^{n/2} det Lambda = gamma_n^{n/2}$ (in your case), $gamma_n$ being the $n$-th Hermite constant, whence you get $A le prod_{i=1}^n |b_i|_2 le frac{gamma_n^{n/2}}{2^n} prod_{i=1}^n sqrt{i + 3}$.