Is it permissible for me to invite a ahl al-kitab woman to the mosque for the purposes of drawing her heart towards Islam and possibly marriage?
There is a big difference between a masjid (mosque) and an Islamic center. The masjid is meant only for Muslims. On the other hand, in general, you can invite non-Muslims to visit an Islamic center and give them a tour, but try to avoid the area that is specified for the prayers (i.e. masjid) if possible.