Consider the "generalised Cantor space", the carrier set being $2^{\omega_{1}}$ and the basic open sets being sets consisting of all bit-strings of length $\omega_{1}$ with a fixed countable well-ordered bit-string as initial fragment. Denote this space by $A'$. Consider space with same carrier set with product topology obtained by viewing it as a product of $\aleph_1$ discrete spaces of cardinality two. Denote this space by $A''$.

I define a space $X$ to be a ``nice space" if:

First, the space $X$ is compact and contractible and is the image of $A''$ under a continuous surjection. Secondly, the space $X$ is the disjoint union of an open dense subset $U$ and the complement $V$, and both $U$ and $V$ admit a ``homogenous" metric, where what we mean by this is as follows. Firstly, with regard to $U$, there exists some $\epsilon>0$, with the property that, for all $\delta$ such that $0<\delta<\epsilon$, every pair of distinct open balls of radius $\delta$ centred at a point in $U$ such that the closure of the balls does not intersect $V$, has the property that the balls in the pair are isometric. Then, with regard to $V$, we require that sufficiently small open balls in $X$, centred at points of $V$, of the same radius, are isometric.

Every closed ball in a finite-dimensional Euclidean space is indeed a nice space. The space $A''$ is a $k$-space, and so is every nice space, so exponential topologies exist here. I claim that for each nice space $X$ a continuous surjection $A' \rightarrow X^{A''}$ exists with the following property. Take an open subset of $X$ and take the pre-image under the evaluation map in $A'' \times X^{A''}$, then take projection onto the first factor. The result subset of $A''$ is open in $A''$, and its image under the continuous map $A'' \rightarrow A'$ which fixes every point is open in $A'$. This is enough to apply the method of proof of Lawvere to obtain that every continuous endomorphism of $X$ has a fixed point. Thus Brouwer can be recovered as a corollary of a suitable generalisation of Lawvere.

The question of the existence of a space $A$ with a continuous surjection $A \rightarrow X^{A}$ for every nice space $X$, or for every space $X$ in some class which includes every closed ball in a finite-dimensional Euclidean space, still remains an open problem.

Brief outline of method of proof for showing that the continuous surjection exists.

It is possible to construct an $\omega$-sequence $S:=\{C_{n}:n \in \omega\}$ of coverings of $X$ by finitely many open balls, each covering $C_{n} \in S$ being such that it can be partitioned into two collections of open balls with each collection having the property that all of the balls in it are pairwise isometric, and also such that the mesh of the covering $C_{n}$ tends towards zero as $n$ goes to infinity. Let $T$ be the collection of all centres of open balls occurring in some covering $C_{n}$.

An element of $X^{A''}$ can be coded for, non-uniquely, by a mapping from a countable collection of countable bit-strings $B$, closed under taking initial fragments, and such that every element of $2^{\omega_1}$ has some element of $B$, maximal under the "initial fragment" relation, as an initial fragment, into $X$, with bit-strings of successor length being mapped to elements of $T$.

One can further require that the trace of the mapping into $X$ along each branch of $B$ is ``generalised Cauchy"; it satisfies an obvious generalisation of the Cauchy criterion for each fragment of the branch of limit length, relative to the metric on $X$ which we have been holding fixed throughout, and with the speed of convergence having a uniform lower bound across all fragments of branches of $B$ of a given limit length.

Consider set of all mappings from such a set $B$ into $X$ satisfying the ``generalised Cauchy criterion" in question; this is in one-to-one correspondence with an appropriate set of countable well-ordered bit-strings $D$ under an appropriate coding scheme. Every element of $X^{A''}$ is coded for by at least one such mapping which in turn can be coded for by a countable well-ordered bit-string; further argument is required to show that every countable well-ordered bit-string does indeed code for an element of $X^{A''}$. To show that part of it, need to use a transfinite induction argument showing that from every mapping from an appropriate set $B$ into $X$ satisfying the appropriate constraints we can recover a continuous map $2^{\alpha} \rightarrow X$ for each countable ordinal $\alpha$ ($2^{\alpha}$ having the product topology), and that when $\alpha$ is sufficiently large, this map $2^{\alpha} \rightarrow X$ determines the map $2^{\beta} \rightarrow X$ (also continuous) for all $\beta$ such that $\alpha \leq \beta \leq \omega_1$. To get this transfinite induction to work we need to use both the "generalised Cauchy criterion" which is assumed to hold for our map $B \rightarrow X$ and also the given hypotheses on the space $X$, including the existence of an appropriate kind of metric on $X$.

Then, having done that, you need to show that the coding scheme for such maps $B \rightarrow X$, each such map being represented by a countable well-ordered bit-string from $D$, can indeed be constructed in such a way that it induces a surjection $A' \rightarrow X^{A''}$, continuous relative to generalised Cantor space topology on $A'$, and exponential topology on $X^{A''}$ arising from product topology on $A''$, which does indeed have all the properties I claimed.

That in very brief outline is the proof. Perhaps this post will still be thought inappropriate since I have not given all details. Or on the other hand maybe it can be accepted as a partial answer to the question.