fedja: This is a very nice and elegant answer, thank you! (Have you seen/used such a norm ever before?) Before accepting your answer, I'd like to wait a bit, so that other participants feel more encouraged to present other answers.

In fact, let me give here a modification of your answer. I noticed that for $N=3$ your norm of $v=(v_1,\dots,v_N)$ is $\sum_i|v_i-\frac12\,\sum_1^N v_j|$. So, I thought such a modified norm will work as well, for all $N\ge3$, perhaps with some other factor in place of $\frac12$. This does not seem to work, though. [**Addendum:** Of course, since the conjectured inequality in my question holds when $B$ is one-dimensional, it was silly for me to think even for a moment that such an $\ell_1$ norm as $\|v\|:=\sum_i|v_i-\frac12\,\sum_1^N v_j|$ could possibly provide a counterexample.]

However, if I am not mistaken, if $\|v\|:=\max_i|v_i-\frac14\,\sum_1^N v_j|$ for $v=(v_1,\dots,v_N)$, then
$\|e_1\|=3/4$, $\|e_1+e_2\|=1/2$, and $\|e_1-e_2\|=1$, so that your main inequality $(*)$ holds for all $N\ge5$.

Only the dimensions $N=3,4$ now remain to be considered.