I hope your mother has great eyes! Working with black can be challenging for some of us.
I checked out the link provided in the original post. CW is ***so*** much work, I would not risk using such an inexpensive, no-name broadcloth. Without actually trying it out, you have no idea what its actual color is (there are washed-out looking blacks), how much color it will retain after washing (some blacks lose depth), how it will shrink (non-brand-name cottons can have weird shrinking and wrinkling characteristics), how it will wear, etc. For a CW quilt, you want reasonably high-quality fabric. There are high-quality muslins on the market, but they do not come in black. All blacks you will find will be a broadcloth weave. I don't know why this is, but it is what I have found.
There is no guarantee with any brand that blacks (or any other color) purchased at different times will match exactly. Often they will vary somewhat in tone and color. I would definitely purchase all of the backing fabric at the same time, and over-buy in case there are some mistakes before the quilt is finished.
I agree with using a name-brand black.
Michael Miller Jet Black is my favorite black because it is deeper and darker than other blacks and it has a very nice hand.
https://www.fabric.com/buy/ej-764/mi...ller-jet-black When compared side-by-side with other blacks, MM JB always wins.
Kaufman Kona tends to have a "meatier" hand than I like for quilts. That is, if you work with it, it feels thicker and heavier than other brand solids. Personally, I don't like it, but it may be exactly what your mother is looking for. The hand of Moda Bella is more like other good quality quilting fabrics but, as I said, I prefer the Michael Miller Jet Black for its deep color.
Is your mother planning to make this quilt by hand or by machine? Some of the above fabrics are probably easier to work with by hand than others. What you might want to do is buy small amounts of several blacks for your mother to try out, then let her decide which one she wants to use.