Upper bound: $S_m(k,\ell) \le m/\gcd(m,\ell-1) - 1$

This is the upper half of the closed form, proved at the level of residues.

Throughout, $m \ge 2$ is the modulus, $\ell \ge 2$ the number of summands, $k \ge 1$ the number of colour classes, $d = \gcd(m, \ell - 1)$ and $n = m/d$.

For every $m \ge 2$, $\ell \ge 2$ and every number of colours $k$,

No hypothesis on $k$ is needed: the bound holds uniformly, however many colours are available. The reason is that once the interval reaches $n$, whichever class receives the residue $\overline{n}$ fails to be $\ell$-sum-free, and adding colours does not help.

Uniformity in $k$ is what makes this the ceiling of the whole problem, and it is half of the closed form: the matching lower bound only needs enough colours to be available.