CG spaces from the perspective of sheaves over compact Hausdorff spaces

The following paper seems worth looking at, although I am not certain whether the proofs therein are related to yours:

Martín Escardó, Jimmie Lawson, Alex Simpson: Comparing Cartesian closed categories of (core) compactly generated spaces. Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145.

The evil link and the good link are both available.

I found an answer using Ivan Di Liberti's very helpful reference in the comments. I figured I would post it here in case anyone has interest in this approach to compactly generated spaces in the future.

For a cardinal $\kappa$, let $I_{\kappa}$ be the category of compact hausdorff spaces with cardinality less than or equal to $\kappa$. The cardinal is to deal with size issues. We make $I_{\kappa}$ into a site where covers are jointly surjective collections of maps. $I_{\kappa}$ induces an adjunction between $\text{Sh}( I_{\kappa}^{op})$, the category of sheaves of sets on $I_{\kappa}^{op}$, and $\text{Top}$. The left adjoint $L$ sends $F$ to $\int_{K \in I_{\kappa}} F(K) \times K$, and the right adjoint $R$ sends $X$ to the sheaf sending $K$ to $[K, X]_{\text{Top}}$. $L$ is a left Kan extension of $I_{\kappa} \rightarrow \text{Top}$ along the inclusion $I_{\kappa} \rightarrow \text{Sh}( I_{\kappa}^{op})$.

This adjunction between $L$ and $R$ is idempotent; it factors as $\text{Sh}( I_{\kappa}^{op}) \leftrightarrow \text{CG}_{\kappa} \leftrightarrow \text{Top}$, where $\text{CG}$ is what I call $\kappa$-compactly generated spaces. $\text{CG}_{\kappa}$ is a reflexive subcategory of $\text{Sh}( I_{\kappa}^{op})$ and a coreflexive subcategory of $\text{Top}$. What Ivan Di Liberti realized is that this may allow us to apply proposition 4.3.1 in The Elephant. Let $L' : \text{Sh}( I_{\kappa}^{op}) \leftrightarrow \text{CG} : R'$ be the reflexive adjunction. To show that $\text{CG}_{\kappa}$ inherits a cartesian closed structure from $\text{Sh}(I_{\kappa}^{op})$, it suffices to show that $L'$ preserves products.

$L'$ sends a representable sheaf $[-, K]$ to $K$. It preserves products of compact hausdorff spaces; the canonical map $L'([-, K] \times [-, K']) \rightarrow L'([-, K]) \times L'([-, K'])$ is an isomorphism. To see this, note that it is a bijection (one can show that it is an isomorphism after composing with the forgetful functor to set, and this can be made very canonical). Now the famous lemma about a continuous map from a compact space to a hausdorff space being open finishes the job.

This result can be extended to work for the general case. $\text{Sh}(I^{op}_{\kappa})$ is a sheaf topos. It is therefore locally presentable (see Borceux, Handbook of Categorical Algebra, prop. 3.4.16, page 220). So there is a small set of $\lambda$-small objects which generates this category under $\lambda$-filtered colimits. Every sheaf is a $\lambda$-filtered colimit of representables here. $L'$ and products commute with $\lambda$-filtered colimits, reducing to the previous paragraph.

Now it seems that, if we take the large colimits $\text{colimit } \text{Sh}(I_{\kappa}^{op})$ and $\text{colimit } \text{CG}_{\kappa}$, I think we obtain compactly generated spaces, along with the desired product preserving reflexive adjunction.