If I understand you goal correctly, then will you a select inside the Customizer, there list the Categories and the User can use this for save in your theme settings.
If you will enhance the Customizer for categories, then use this class from this repo here. Include the file and enhacne the default class. You can easy implement a select for the categories of the install. Inside the readme of this repo are two links to gist with a example use, but for other class of the repo. I think is clear and the same way on all extensions for the Customizer.