Optimal Coverage Sets of a Sum of Random Variables with Known Copula

I don't know if the following leads to a "practical" solution but it is a characterization of the minimum possible length. This responds also to the final line in the OP's post,

Additionally, I have a feeling that there should exist some theory describing a set S as a function of copula and marginal distributions $F_X,F_Y$.

Since $S$ is an interval, we can write $S = [\,\underline {s},\, \underline{s}+L]$, where $L>0$ is the length. Also set $Z \equiv X+Y$. Then the required property can be written as

$$\Pr(Z \leq \underline{s}+L) - \Pr(Z \leq \underline{s}) \geq 0.9$$

The two probabilities can now be written in terms of the cumulative distribution function of $Z$, say $F_Z$, so we require

$$F_Z(\underline{s}+L)) - F_Z(\underline{s}) \geq 0.9$$ $$ \implies L \geq F_Z^{-1}\big(0.9 + F_Z(\underline{s}) \big) - \underline{s}.$$

This last expression characterizes the minimum possible length in terms of the CDF and the quantile function of $Z$, and of $\underline{s}$.

We can write the CDF of $Z$ using eventually the copula density of $X,Y$ as

\begin{align} F_Z(z) = \Pr(Z \leq z) &= \Pr(X+Y \leq z) = \Pr(X \leq z-Y) \\ & = \int_{-\infty}^{\infty} f_Y(y)\int_{-\infty}^{z-y}f_{X|Y}(x|y) dx dy\\ & =\int_{-\infty}^{\infty} \int_{-\infty}^{z-y}f_Y(y)f_X(x) c\Big(F_X(x), F_Y(y)\Big) dx dy \end{align}

One perhaps could also exploit the other properties stated in the OP's post, related to the marginals of $X,Y$ to arrive at something more specific.