Ok, there is a lot to say here, but I should first of all say that I am not really an expert. I think that there has been some implementation of nonstandard analysis in the Wolfram language, although I cannot remember where.
About using Wolfram language, you can get started without a license at the Wolfram Open Cloud.
About Chu spaces, there is a concrete definition given on page 4 of the Samson Abramsky paper that your FQXI paper references. There are two sets and an evaluation function from their Tuples ( a third set is implicit as the range of that function).
What I recommend is to start with the simplest case. Maybe have X, A, and K be two element sets and then find all e that satisfy the axioms. There are many ways to do functions in Wolfram language, but you'd want to pick a way that makes it easy to enumerate, e.g. using an array of values. You will want a way to visualize it. Having a multiplication table (like in your essay) is a fine thing. This could all be done using functions like Table for enumeration, and Select for picking out cases with the right properties, and ArrayPlot.
This follows a general scheme inspired by Wolfram's book. Enumerate the simplest cases and then see what they do.
In terms of the interpretation, it does not matter that it is too simple for what you want. The point is to be objective and you have to understand the simplest case first. Going to three element sets is not unreasonable.
In the further case that you want X to be a space and A to be a collection of sets in that space (maybe closed under intersection) that is also something you can enumerate. Then you can make more elaborate visualizations if you have one for X.