Noam Elkies pointed out to me by email that there cannot exist a dissection (of the type I asked for) using bounded number of pieces. Consider a triangle with sides $T+1$, $T+1$, $2T$, so $s=2T+1$ and $r\approx \sqrt{T/2}.$ Then we are trying to dissect a cuboid $X$ with sides $T$, $T$, $1$ into a cuboid $Y$ with sides $\sqrt{T/2}$, $\sqrt{T/2}$, $2T+1$. It is then intuitively clear that since $X$ has thickness $1$, it is "too thin" relative to $Y$ for any dissection into $C$ pieces—where $C$ is a constant independent of $T$—to work. (A rigorous proof is a bit tricky; it's an <a href="https://math.stackexchange.com/q/231051">old chestnut</a> to show that a disc of diameter $D$ cannot be covered with fewer than $D$ rectangles of width $1$.)
On the other hand, it's still conceivable that by adding some auxiliary pieces $Z$, there might be a scissors congruence between $X+Z$ and $Y+Z$ using a bounded number of pieces.