My suggestion would be that, since you want a tiny integer-only NN tailored for a specific computer, are only occasionally training one for a specific task, and you have a simulator to generate unlimited data, you simply do random search or an evolutionary method like CMA-ES.

They are easy to understand and code up by hand in a few lines (which is one reason you won't find any libraries for them - they are the 'leftpad' or 'isEven' of NNs, the effort it would take to install and understand and use a library often exceeds what it would take to just write it yourself), will handle any NN topology or numeric type you can invent, and will train very fast in this scenario.