Thanks for the detailed feedback! Testing with a single controller plugged in it seems fine. I am aware there is an issue if you have multiple controllers plugged in. Do you maybe have more than one controller connected?
I honestly have no idea what to answer you, because I only have one Xbox One controller (wired) and the keyboard connected. I don't know if I have drivers that make Retrofab think it's receiving inputs from multiple sources, although this shouldn't be an issue. I imagine it's not very helpful, but I can also replicate this error in the browser. In any case, I can play without problems if I don't assign Mode buttons to the controller. I just wanted to warn that this issue exists, even though it might only happen to a few people.
As a side note, I haven't done exhaustive testing, but this happens with games that have configurable Game, Game A and/or Game B buttons, and that don't have these Game buttons assigned to controller buttons by default. It happens with practically all Game & Watch games and, for example, with Elektronika clones.