Add Custom Attributes To Product Filters in Widget Setup

I’m trying to add custom columns / attributes to the product filter when configuring a new widget.

I’ve narrowed it down to
magentomodule-catalogviewadminhtmltemplatescatalogproductwidgetchooserchooser.php

but I can’t quite get it working properly.

Ideally, I’d like to a couple of attributes but in addition, I’d like to filter on Attribute Set but as that isn’t an attribute, I’m a bit stuck.

As an example, the code that returns values in chooser.php is

$this->addColumn(
        'chooser_name',
        [
            'header' => __('Product'),
            'name' => 'chooser_name',
            'index' => 'name',
            'header_css_class' => 'col-product',
            'column_css_class' => 'col-product'
        ]
    );

The section in the admin that shows the columns presents them like the following but it’s difficult to manage when you’ve only got ID, SKU & Name as columns.

Would appreciate your help.