This is, indeed, a fairly natural property that is probably the best we can get, and it is very strong. As you point out, for $S = \mathbb{N}$, it implies LPO. (Proof: consider $(b_n)$ a binary sequence at most one term of which equals $1$, let $I_n$ be the open interval from $n-\frac{3}{4}$ to $n+\frac{3}{4}$ except that $I_0$ starts at $-∞$; let $U$ be the union of the $I_n$ such that $b_n=0$ and $V$ be the union of those for which $b_n=1$; then $\mathbb{R} ⊆ U ∪ V = \bigcup_n I_n$; but if $\mathbb{N} ⊆ U$ then $∀n. b_n=0$ whereas if $\mathbb{N} ∩ V$ is inhabited then $∃n. b_n=1$. ∎)
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