How to choose between different types of feature scaling?
You've listed a few different error metrics, I would pick one metric that is best suited to your problem. Trying to maximize several metrics at once makes it difficult to tell if your model is getting better. In any case, if normalization leads to the best score - then that is your answer. Since all other variables were already in range [0,1], then that's probably what I would have started with just to keep it consistent.
I am curious how much worse the z-score standardization is performing compared to normalization. From what you've written about the data - I'd be surprised if standardization vs. normalization produced significantly different results when only 3 out of 45 features are impacted.
This blog explains the topic better than I can About Feature Scaling. This article has several examples of training a MLP with scaled and unscaled variables How to use Data Scaling Improve Deep Learning Model Stability and Performance.
I'd also consider some simpler algorithms. It might be good to also train a logistic regression or similar as a comparison, fewer parameters will be easier to configure and less prone to error. A neural net will give you a boost if you have a lot of training data or a very non-linear problem.