re: Is there a way that a WP user at the level of Contributor could use the Export Tool?
You could use the Members plugin to add the necessar capabilites in or to perform exports. I think this is the complete list of capabilities you’ll need to add:
It looks like the Contributor role already has the
edit_posts capability so just adding the
export capability should do it.
re: If yes, is there a way to add the Export button from Connections Pro onto a forward-facing (but private) page on the site?
Sorry, no this is not possible. The code can not be run from the site’s frontend.
Hope this helps!