Finding a perfect complex with specified support

Question

Let $X$ be a quasicompact quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Then there exists a perfect complex $F \in D^{\mathrm{perf}}(X)$ with $\operatorname{supp} F = Z$. The proof of this in a paper I'm reading uses the induction principle for qcqs schemes and the following lemma: given $X, Z$ as above and any open set $U$, if $F$ is a perfect complex on $U$ with support contained in $Z\cap U$ then there is a perfect complex $G$ on $X$ with support contained in $Z$ which restricts to $F \oplus \Sigma F$ on $U$. They deduce this from a more general result saying there's a $G$ restricting to $F$ iff the class of $F$ is in the image of the map $K_0(X, Z) \to K_0(U, Z \cap U)$ (and then easily $F \oplus \Sigma F$ has class zero, so is in the image). I don't know anything about K theory and it seems like overkill here. Is there a more direct argument, either for producing the perfect complex $F$ or for proving that a perfect complex with euler characteristic zero can be extended to the whole space (still supported in the right place)? I guess this is the opposite of a direct argument, but I'd also be happy if there was some cleverness we could do using descent for $D^{\mathrm{perf}}$.