Question regarding Tag Widget

On our website we are using the native WP search to search the blog articles and the Q&A search for everything Q&A related, for this we are using $wp_post_types['question']->exclude_from_search = true;. But if $wp_post_types['question']->exclude_from_search = true; and I click on a Tag on the Tag Widget, I will get an page not found message. And if $wp_post_types['question']->exclude_from_search = false; the Tag Widget will work as it should but WP search will display as results blog articles and Q&A posts. Is there any away around this, where I can still keep the WP search only returning the blog post and Q&A search retuning Q&A posts?

Thank you in advance for you help!