Scaling the data iteratively one by one vs batch scaling
With the first approach, you are completely disregarding the global scale of the signal and only focusing on the relative scale. This will most probably hurt the performance of whatever system you train on that data (or any analysis you perform on it), compared to training on the globally scaled data, unless the relative values are the only important pieces of information in your signals.
I noticed that you already know the range in which the data is defined: (-127, 128). If you want to scale your data, why don't you scale the data with a fixed computation like $((x + 127) \cdot 2 / 255) - 1$ ?
Anyway, I don't see how a mere linear change in scale would save memory or help visualization, which seem to be your final goal.