I actually fixed it and got it working. Turns out I had to configure the WAN interface on the host side.
My /etc/network/interaces file now looks like this:
# This file describes the network interfaces available on your system
# and how to activate them. For more information, see interfaces(5).
# The loopback network interface
auto lo
iface lo inet loopback
# The primary network interface
auto eth0
iface eth0 inet static
address 0.0.0.0
auto eth0:1
iface eth0:1 inet static
address 192.168.0.2
netmask 255.255.255.0
gateway 192.168.0.1
dns-nameservers 192.168.0.1
# The WAN interface
auto eth1
iface eth1 inet static
address 0.0.0.0