It seems like mostly a specific language (English) problem. In German language this would be much less problematic. The term 'spaltenanzahl' is not a strange word and regularly used. So you may consider introducing 'column-count' or 'column count' (while column-number would be ambiguous), or accept 'number of columns' as not that complex after all (and can often be written simpler with other sentence constructions e.g. 'matrix $M$ has $w$ columns').
Some background on the article and examples of sentences would help to look for different terms. One could use width and length or (as Sylvester, origin of the term matrix, did use) breadth and depth. Maybe based on what the matrix actually presents (e.g. system of equations, a polynomial, a vector space, etc.) other terms could be used.
Depending on your article background (or maybe no, independent from the public for your article, maybe only if you do something entirely new) I would advise to not use any newly invented term, and neither use some existing term (which must be an archaic term).
You have to ask yourself whether the poverty in the English language, not containing a simple synonym for the German term spaltenanzahl, is worth introducing something fancy that may only be confusing.