Solved with ETS1331, amuthup, AI216, squareman. The answer is all $S \leq 152$. Claim: If $S$ works, then $S-1$ works. Proof: Take a set summing to $S-1$. Add a copy of $1$. Then there exists a partition into two sets summing to at most $80$ each, and removing the copy of $1$ still leaves us with a valid construction. Note that the contrapositive of the claim also holds. In particular, it suffices to show that $152$ works but $153$ does not. First, suppose FTSOC that some set does not work for $S = 152$. This set contains less than $8$ copies of $10$, as otherwise we would be done. But then we can just add numbers until their sum is between $72$ and $80$, inclusive, at which point we're also done. To show $153$ doesn't work, simply consider the set $\{9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9\}$. It is clear that at least one set has $9$ 9s, implying that it has sum $81$, contradiction.