When I posted the question, I thought there might be an algorithm, but now I think the answer is no.
If it were possible to solve this problem in $o(3^{n/2})$ time in the worst-case scenario via a deterministic algorithm, it would be possible to manipulate the equation $\sum_{i \in I}s_i - \sum_{j \in J} s_j=t$ so that there are $L$ possible expressions on the left-hand-side and $R$ possible expressions on the right-hand-side, where $L+R=o(3^{n/2})$. But this appears to be impossible.
The best you can do is manipulate the equation to get $\Theta(3^{n/2})$ possible expressions on the left-hand-side and get $\Theta(3^{n/2})$ possible expressions on the right-hand-side, as follows:
$\sum_{i \in I^+}s_i - \sum_{j \in J^+} s_j=t-(\sum_{i \in I^-}s_i - \sum_{j \in J^-} s_j)$,
where $I^+ =\{1,\dots,[n/2]\} \cap I$, $I^- =\{[n/2]+1,\dots,n\} \cap I$, $J^+ =\{1,\dots,[n/2]\} \cap J$, and $J^- =\{[n/2]+1,\dots,n\} \cap J$. This allows one to apply the Meet-in-the-Middle algorithm. I don't think anyone will find any algorithm that beats the Meet-in-the-Middle for this problem.
Added in 2019: this paper came out, which gives a probabilistic algorithm and also has a better name for the problem, "equal subset-sum". https://arxiv.org/abs/1905.02424