Capacity of Uniquely Solvable Puzzle (USP)
Like many other questions, the answer to this one can be found in Stothers' thesis. A local USP is a CWP in which the only way in which a 1-piece, a 2-piece and a 3-piece can fit together is if their union is in $S$. Clearly a local USP is a USP, and a construction from [CKSU] shows that the USP capacity is achieved by local USPs (we are going to show that constructively).

Coppersmith and Winograd construct an almost 2-wise independent distribution $S$ on $2^V$ with the following two properties: (1) $\Pr[x \in S] = (|V|/2|E|)^{1-\epsilon}$, (2) For any $x,y,z \in V$ such that the 1-piece of $x$, the 2-piece of $y$ and the 3-piece of $z$ together form a vector $w \in V$: if $x,y,z \in S$ then $w \in S$.

We choose a random subset $S$ of $V$ according to the distribution, and for each edge $(x,y) \in E$, we remove both vertices $x,y$. The expected number of vertices left is roughly $(|V|^2/2|E|)^{1-\epsilon}$. The resulting set $T$ is a local USP: if there are $x,y,z \in T$ in which the 1-piece of $x$, the 2-piece of $y$ and the 3-piece of $z$ fit, forming a piece $w$, then $x,y,z,w \in S$, and so all of $x,y,z$ are removed from $S$.