Sounds like a situation I'm facing: assisting a non-profit org that wants a new website. Some members push for using Wordpress as the CMS. However the org has an established pgsql db containing lots of data that optimally could be employed on the site, e.g., lists of members/events, searchable documents, etc.
Also, there's a separate web app for accessing the database using SQL features mysql doesn't support. Lack of Wordpress/pgsql compatibility makes integrating the existing db and folding in the webapp functionality just about impossible, at least I'm not seeing how that could be done.
Since Wordpress isn't going to embrace pgsql, I've recommended considering alternative CMSs. Decisions are still pending.