We should have the link to create a bug for Bugzilla, instead of linking to the normal bug create page with the product set, we should use a custom template exposing only those fields we are interested in and with a note pointing out this is for Bugzilla itself and not one of the products hosted on Bugzilla.

e.g.
https://bugzilla.redhat.com/enter_bug.cgi?product=Fedora&format=fedora-review

Which comes from extensions/RedHat/template/en/default/bug/create/create-fedora-review.html.tmpl