It sounds like the CS minor would be a worthwhile addition so long as money permits. HTML/CSS/JS will be the bulk of your work as a front-end web developer, and the CS knowledge will help write better code, particularly when it comes to JS. It sounds like the CS major would be a bit much. Your degree should be sufficient - people will care much more about your skills, which you will also help through the minor. But if it would be too expensive or you do not feel the time is worth it, I think you would be okay without it in the worst case.
Good luck!