Apologies if this won't be an answer, but I can't put much more headspace in this question other than brainstorm in public. Let's see if we can reverse engineer the definition!

Coarse structures in a nutshell:

Fix a set $X$; a coarse structure on $X$ collection of relations $\cal E$ such that

- the diagonal $\Delta$ is in $\cal E$;
- $\cal E$ is closed under composition of relation;
- $\cal E$ is downward closed;
- $\cal E$ is closed under taking inverse relation and union of relations.

Monoidal topology in a nutshell:

Fix a monad $T$ on $Set$, a quantale $\cal V$ and define, for each *lax lifting* of $T$ to ${\cal V}\text{-}Rel$ (the usual category of relations, just "V-valued"), called $\hat T$, a bicategory $(\hat T,V)\text{-}Rel$ where objects are sets, and a 1-cell $X\to Y$ is a function $TX\times Y \to \cal V$.

For suitable choices of T and V, a reflexive and transitive $(\hat T,V)$-relation is one of the spaces you want to talk about.

> What could T and V be, in order to describe coarse spaces?

We can try taking as $\cal V$ just {0,1}; then, $\cal E$ is a subset of $Rel(X,X)$, i.e. a map $E:Rel(X,X)\to 2$, such that

 - $E(id_X)=1$; (condition 1)
 - $E$ is a presheaf $Rel(X,X)\to \{0\le 1\}$; (condition 3)
 - $E$ is a monoid homomorphism; (condition 2? $Rel(X,X)$ is an -ordered- monoid under composition)
 - and in fact I believe one can leverage on the whole structure of $Rel(X,X)$ (Rel is a quantaloid!) to capture "functorially" also the remaining condition.

This is definitely a thing. I hope you find a definitive answer.