Change permissions of postgres copy to exported file
You'll want to create a directory without the t (sticky) permissions bit set, and have PostgreSQL copy the file into there. You should also make the directory setgid (g+s), as that makes the group inherited from the directory, not the creating process; see setgid directories.

You could put this directory in /tmp but personally I'd put it some neutral location like /var/lib, or mounted volume somewhere, as /tmp gets cleaned out automatically so you'll need to have the directory re-created if it's missing.

Assuming you do use /tmp:

mkdir -p /tmp/pgcsv
chgrp users /tmp/pgcsv
chmod u=rwX,g=rwsX,o= /tmp/pgcsv


then create the csv in there. It'll be automatically created with group-owner users, which will make it readable by your users (unless PostgreSQL's umask is too restrictive). Because the directory is group-writable by users (or whatever group you use) and does not have the t (sticky) bit set your users can delete files they do not own from it.

The reason this doesn't work when using /tmp/ directly is that the sticky bit is set in /tmp for security, so that user1 can't delete a file created by user2 and replace it with a symlink that user2 then writes to, allowing user1 to steal their data / trick them into overwriting things. 



Other options:


Run the cron job as user export, using psql's \copy command, so it writes the data over the client/server socket and the file is created as the export user in the first place. This is by far the simplest option, and only slightly slower than a direct COPY.
Have a cron job running as root invoke the PostgreSQL copy then move the data file and change its ownership once the COPY finishes.
Give the export user the ability to run a very restricted command as root via sudo to move/delete.
Use POSIX ACLs to grant the postgres user special permission to write to /home/export (if POSIX ACLs are enabled on your file system)
Set group-owner of /home/export to postgres and make its permissions g=wx. PostgreSQL can add new entries, and overwrite anything within that it has write permission to, but not list entries. I don't like this option much.