However, because topology on $A^{\prime }$ is finer than topology on $A^{\prime \prime }$ here, this still shows how the proof of the Lawvere fixed point theorem can be applied here to give Brouwer fixed point theorem as corollary, which could conceivably be a publishable result (see what "Geometry and Topology" think about that), and this could still be sorta kinda maybe relevant to Scott's original motivation for looking at the problem (if you're okay with working with two different topologies on the space of agents, one finer than the other). But this is a very big space of agents you're talking about here.

Correction: need not only that topology on $A^{\prime }$ is finer than topology on $A^{\prime \prime }$, but also, given arbitrary open subset of $X$, take pre-image under evaluation map in $X^{A^{\prime \prime }}\times A^{\prime \prime }$, projection onto first factor and then pre-image of that under the continuous surjection $A^{\prime }\to X^{A^{\prime \prime }}$, it needs to be shown that this set is open in both topologies. I believe that this can indeed be done for an appropriate class of spaces $X$ for the pair of topologies in question.

Let $A^{\prime }$ be $2^{{\omega }_1}$ with generalised Cantor space topology, and $A^{\prime \prime }$ be $2^{{\omega }_1}$ with product topology, $X$ a closed disc in a finite-dimensional Euclidean space. Then there is a continuous surjection $A^{\prime }\to X^{A^{\prime \prime }}$. I don't know how to show that there is a topological space $A$ with carrier set $2^{{\omega }_1}$ and a continuous surjection $A\to X^A$. Thanks to Alex Mennen for pointing out the problem.

When I look at my post the LaTeX code isn't formatting properly; if anyone can let me know how to fix that.

I have just now submitted an attempted solution to this problem to "Geometry and Topology". I claim that the space $X$ you are looking for is $2^{{\omega }_1}$ (${\omega }_1$ being the least uncountable cardinal) with the ``generalised Cantor space topology", that is for each countable well-ordered bit-string $b$ you have a basic open set consisting of all bit-strings of length ${\omega }_1$ with $b$ as an initial fragment. Since this topological space has quite a large cardinality I'm somewhat unclear whether this is helpful for your proposed application and would need to think about it more. (Matthew Barnett just now directed me to this post of yours.) I sent you an early draft of my paper, which argues the point in detail, on FB Messenger, and can send the latest version to you if you wish.