Reading pingviin.org forum, mint 13 is reported to work fine with RIA ID-card software, so I really doubt that changing distro would help. Core problem must be somewhere else, but if you insist changing a distro, then I would recommend ubuntu or mageia. Ubuntu is directly supported by RIA and mageia has esteid in its repos and is installed when you choose Estonian language during installation process.
I use Fedora and I'm currently in process of pushing that software into Fedora repos. It is going SLOW :)
Meanwhile I provide copr repo (like PPA in ubuntu)
http://copr.fedoraproject.org/coprs/mihkel/esteid/
You can install xfce in any distro.
BUT, as I already mentioned, I don't think that changing a distro helps. If I were you, I'd borrow another reader from someone first and then try other distros.