Well........ I've figured it out. Here's what I discovered:
I had previously converted the texture to 1-bit using the Threshold adjustment in Photo and saved it accordingly. I was then using that same file every time and placing it into the Designer document. Once imported, I could clearly see that it was still 1-bite. However, every time that I rasterized it to mask, the anti-aliasing would occur and nothing could solve that.
Just now, I decided to import the original file of that texture (before the threshold adjustment was applied), then applied the Threshold directly within Designer itself. It converted to 1-bit, of course, and then I rasterized to mask as before. This time, it worked the way it should without anti-aliasing! I then tested with several other similar texture files to verify and it continues to be the case.
For some odd reason, Designer is adding anti-aliasing to any 1-bit textures that are imported and rasterized. However, when the threshold is applied within Designer itself and then rasterized, the anti-aliasing doesn't occur.
Couldn't begin to tell you why, but plenty happy that I can move forward with my work, understanding how to solve the problem now.