Best How To :

There are two ways to achieve this, it just depends on whether you want to edit single-product.php or content-single-product.php. Both will get you to the same end result.

To do the former, you can use default WordPress functionality and filter the template under given conditions via template_include

add_filter( 'template_include', 'so_29984159_custom_category_template' );
function so_29984159_custom_category_template( $template ){
    if ( is_single() && get_post_type() == 'product' && has_term( 'custom_category', 'product_cat' ) ) {
        $template = locate_template( 'single-product-custom.php' );
    }
    return $template;
}

Or if you'd rather create a custom content-product.php template part, you can use WooCommerce's wc_get_template_part which works in pretty much the same way. Technically I think you can use the same conditional logic, but I tweaked it here to point out the variables WooCommerce makes available at the filter.

add_filter( 'wc_get_template_part', 'so_29984159_custom_content_template_part', 10, 3 );
function so_29984159_custom_content_template_part( $template, $slug, $name ){
    if ( $slug == 'content' && $name == 'product' && has_term( 'custom_category', 'product_cat' ) ) {
        $template = locate_template( 'content-product-custom.php' );
    }
    return $template;
}

In either case the single-product-custom.php or content-product-custom.php would be in your theme's root folder. To put them somewhere else, just change the path in the locate_template() argument.