The question is \'etale local on $X$ hence we may assume $X$ is affine. Say $X = \Spec(A)$. Then we have $D_\QCoh(\mathcal{O}_X) = D(A)$ by Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-compare-bounded}. Hence we can choose an K-flat complex of $A$-modules $K^\bullet$ whose corresponding complex $\mathcal{K}^\bullet$ of quasi-coherent $\mathcal{O}_X$-modules represents $K$. We claim that $\mathcal{K}^\bullet$ is a K-flat complex of $\mathcal{O}_X$-modules. \medskip\noindent Proof of the claim. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-K-flat} we see that $\widetilde{K}^\bullet$ is K-flat on the scheme $(\Spec(A), \mathcal{O}_{\Spec(A)})$. Next, note that $\mathcal{K}^\bullet = \epsilon^*\widetilde{K}^\bullet$ where $\epsilon$ is as in Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site} whence $\mathcal{K}^\bullet$ is K-flat by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-K-flat-points} and the fact that the \'etale site of a scheme has enough points (\'Etale Cohomology, Remarks \ref{etale-cohomology-remarks-enough-points}). \medskip\noindent By the claim we see that $La_X^*K = a_X^*\mathcal{K}^\bullet$ and $L\pi_X^*K = \pi_X^*\mathcal{K}^\bullet$. Since the first part of the proof shows that the pullback $a_X^*\mathcal{K}^n$ of the quasi-coherent module is acyclic for $\epsilon_{X, *}$, resp.\ $a_{X, *}$, surely the proof is done by Leray's acyclicity lemma? Actually..., no because Leray's acyclicity lemma only applies to bounded below complexes. However, in the next paragraph we will show the result does follow from the bounded below case because our complex is the derived limit of bounded below complexes of quasi-coherent modules. \medskip\noindent The cohomology sheaves of $\pi_X^*\mathcal{K}^\bullet$ and $a_X^*\mathcal{K}^\bullet$ have vanishing higher cohomology groups over affine objects of $(\textit{Spaces}/X)_\etale$ by Lemma \ref{lemma-vanishing-adequate}. Therefore we have $$ L\pi_X^*K = R\lim \tau_{\geq -n}(L\pi_X^*K) \quad\text{and}\quad La_X^*K = R\lim \tau_{\geq -n}(La_X^*K) $$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-is-limit-dimension}. \medskip\noindent Proof of $L\pi_X^*K = R\epsilon_{X, *}(La_X^*\mathcal{F})$. By the above we have $$ R\epsilon_{X, *}La_X^*K = R\lim R\epsilon_{X, *}(\tau_{\geq -n}(La_X^*K)) $$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Rf-commutes-with-Rlim}. Note that $\tau_{\geq -n}(La_X^*K)$ is represented by $\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)$ which may not be the same as $a_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)$. But clearly the systems $$ \{\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)\}_{n \geq 1} \quad\text{and}\quad \{a_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)\}_{n \geq 1} $$ are isomorphic as pro-systems. By Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) and the first part of the lemma we see that $$ R\epsilon_{X, *}(a_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)) = \pi_X^*(\tau_{\geq -n}\mathcal{K}^\bullet) $$ Then we can use that the systems $$ \{\tau_{\geq -n}(\pi_X^*\mathcal{K}^\bullet)\}_{n \geq 1} \quad\text{and}\quad \{\pi_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)\}_{n \geq 1} $$ are isomorphic as pro-systems. Finally, we put everything together as follows \begin{align*} R\epsilon_{X, *}La_X^*K & = R\epsilon_{X, *} (R\lim \tau_{\geq -n}(La_X^*K)) \\ & = R\lim R\epsilon_{X, *}(\tau_{\geq -n}(La_X^*K)) \\ & = R\lim R\epsilon_{X, *}(\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)) \\ & = R\lim R\epsilon_{X, *}(a_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)) \\ & = R\lim \pi_X^*(\tau_{\geq -n}\mathcal{K}^\bullet) \\ & = R\lim \tau_{\geq -n}(\pi_X^*\mathcal{K}^\bullet) \\ & = R\lim \tau_{\geq -n}(L\pi_X^*K) \\ & = L\pi_X^*K \end{align*} Here in equalities four and six we have used that isomorphic pro-systems have the same $R\lim$ (small detail omitted). You can avoid this step by using more about cohomology of the terms of the complex $\tau_{\geq -n}a_X^*\mathcal{K}^\bullet$ proved in Lemma \ref{lemma-vanishing-adequate} as this will prove directly that $R\epsilon_{X, *}(\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)) = \tau_{\geq -n}(\pi_X^*\mathcal{K}^\bullet)$. \medskip\noindent The equality $K = Ra_{X, *}(La_X^*\mathcal{F})$ is proved in exactly the same way using in the final step that $K = R\lim \tau_{\geq -n}K$ by Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-nice-K-injective}.