How to refresh Classes view for active Component in the Style Manager

I'm using a trait (type=select) to change the class of a component (a "column" component, so classes representing width 1-12). After I have removed previous col-* classes and added the newly selected one (via a function triggered by the select change), the Classes section of the Style Manager still has the old class. If I select a different component, then select the column again, it has the new, correct class. How do I force a refresh on the style manager without de/reselecting the component?

Perhaps I am adding/removing classes incorrectly, if the Style Manager view should update automatically. Is this standard Backbone stuff? If so, I'd love to be pointed in the direction of a concise Backbone primer.

Relevant component creation:

var comps = editor.DomComponents;
var defaultType = comps.getType('default');
var defaultModel = defaultType.model;
var defaultView = defaultType.view;

comps.addType('col', {
  model: defaultModel.extend({
    defaults: Object.assign({}, defaultModel.prototype.defaults, {
      'custom-name': 'Column',
      traits: [{
        type: 'select',
        options: /* map of classes */,
        label: 'Width',
        name: 'col-width',
        changeProp: 1
      }]
    }),
    init() {
      this.listenTo(this, 'change:col-width', this.changeColClass);
    },
    changeColClass() {
      var width = this.get('col-width');
      var classes = this.get('classes');

      // remove existing col class
      var col_classes = /* list of classes */
      classes.forEach(element => {
        if (col_classes.includes(element.id)) {
          classes.remove(element);
        }
      });

      // add new class to selector manager and component class list
      var class_name = 'col-' + width;
      var sm = this.sm.get('SelectorManager');
      var class_obj = sm.add(class_name)
      classes.add(class_obj);
    }
  }, {
    isComponent: function(el) { /* check if contains a col class */ }
  }),
  view: defaultView
});