Best way to show relevant experience when applying for a loosely related Masters?
All the usual advice about writing effective CVs and cover letters applies here. Additional advice:


Emphasize the positive and don't explain any negatives, since I really don't see any negatives in your story.
I suggest you take a couple of computer science courses before you apply, in order to (a) get your feet wet, (b) demonstrate your ability to be successful in a graduate CS program, and (c) garner at least one LOR in the target field. You can audit if money is an issue. 


I recommend taking one course per semester if you are working full time. Choose a challenging but doable course that you're truly interested in.