Here is simple construction for $m(n)=2^n$. As is indicated in the remarks above one can't get an essentially lower dimension.

The $E_i$-s will be diagonal matrices. $E_1$ will have the first half diagonal entries equal to 1 the second half 0. $E_2$ the first and third fourths 1-s the rest zeroes, ..., $E_n$ alternatively 1 and zero. (These are $(r_i+1)/2$ where $r_i$ are the Rademacher functions). The point is that for each subset $A$ of ${1,\dots,n}$ there is a $j$ such that ${E_1(j,j),\dots,E_n(j,j)}$ is exactly the indicator function of $A$. The $E_i$ are idempotents.

Using the equivalent $\ell_1$ norm $\|\cdot\|_S$ at the end of the question, it is easy to see that these satisfy the inequalities. They satisfy them with constants 1 for the $\|\cdot\|_S$ norm.

Update: Disregard this answer. It relates to a previous version of Yemon's question.

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Here is simple construction for $m(n)=2^n$. As is indicated in the remarks above one can't get an essentially lower dimension.

The $E_i$-s will be diagonal matrices. $E_1$ will have the first half diagonal entries equal to 1 the second half 0. $E_2$ the first and third fourths 1-s the rest zeroes, ..., $E_n$ alternatively 1 and zero. (These are $(r_i+1)/2$ where $r_i$ are the Rademacher functions). The point is that for each subset $A$ of ${1,\dots,n}$ there is a $j$ such that ${E_1(j,j),\dots,E_n(j,j)}$ is exactly the indicator function of $A$. The $E_i$ are idempotents.

Using the equivalent $\ell_1$ norm $\|\cdot\|_S$ at the end of the question, it is easy to see that these satisfy the inequalities. They satisfy them with constants 1 for the $\|\cdot\|_S$ norm.