Ha. One of the businesses I work with, there is not category for it, it's a new industry that does a variety of things and after exhaustive searches in multiple cities the primary categories of the 3-pack listings are all over the map. I mean like 6 different categories with almost equal distribution. That's why I asked this type of question in the first place.
There is a strong correlation between the search keyword(s) being in the category names. So I would say it is more important to determine your keyword(s) with highest conversion/roi rate and make the closest matching category name to that keyword to your primary. Of course in most cases that will be the one most relevant to your business but not necessarily in all cases.
But it does display the category most relevant to the search. So my strategy would be to make sure each of your primary keywords appears in a category name. How much emphasis Google places on your primary category from the perspective of search result ranking, I don't know but it could be NONE. For all I know the primary category could be just for what they display in your knowledge graph or GMB listing, not in the 3-pack or local finder result, they display the closest matching category to your keyword search there.