@mechy and
@kolla - just wanted to report back to you guys. After a bit of sleuthing around, I did find a Volktek BNC/TP media converter, a couple of T-connectors, a matching pair of 50 Ohm terminators and a matching 50 Ohm coaxial "Cheapernet" cable. After setting everything up and powering on the converter and the Amiga, the converter "Link" LED powers up, suggesting a sound connection between the Ariadne card and the converter.
Unfortunately, that's where my luck ends. Despite having told the Ariadne card to use the BNC jack ("SetEnv Sana2/ariadne_0.config 10BASE2"), despite setting up the card1 ("ifconfig ether0 192.168.1.64 netmask 255.255.255.0"), despite checking the default route ("route add default 192.168.1.1"), AmiTCP still won't detect any incoming data. Pinging localhost works just fine, but pinging the gateway (i.e. 192.168.1.1) ends up with a 100% packet loss. When I ping the router like this, strangely the media converter's "Receive" and "Send" LEDs starting blinking, so there actually seems to be traffic going through the coaxial cable - but for some reason, AmiTCP doesn't see it coming back. Using Roadshow's DHCP functionality doesn't change anything - NetLogViewer shows that the stack never receive an answer from the gateway's DHCP server.
All in all, the outcome is the same no matter if I use the Ariadne's RJ45/Twisted Pair jack, or its BNC connector. Unfortunately, I'm leaning towards the Ariadne being busted somehow - which is really strange. It's been working flawlessly since I got it, and it's been stored in room temperature ever since.