So I think I found out why Blizzard's official SC2 Art Tools were not properly exporting M3 models. The problem happens when I import whatever model into SC2 Map Editor, the model seems to appear just fine, however, no textures work at all.
So after hours of banging my head, I found out why Blizzard's exporter fails, when opening the raw M3 model using Notepad, I discovered that the model exported by SC2 Art Tools is actually telling the SC2 editor to find my textures here:
rather than here:
EDIT: Looks like I was right. After clicking "Move Files" in my Starcraft II import manager, I moved my custom M2 textures to the directory: "Z:\Network" and after doing so, it finally worked!