Since Chiara's onboard ethernet card has a reputation to be flaky in Linux, Koji suggested we could just buy a new ethernet card and throw it in there, since they're cheap.
I've installed a Intel EXPI9301CT ethernet card in Chiara, which detected it without problems. I changed over the network settings in /etc/networking/interfaces to use eth1 instead of eth0, restarted nfs and bind9, and everything looked fine.
Sadly, EPICS/network slowdowns are still happening. :(