Ok - a quick update:
I checked the eth0 configuration via the CLI and it didn't have a default gateway set. I remember trying to set it via the web interface, it kept saying it saved but it would never seem to stick. I configured it with the vami_config_net command from before and reset the network. I'm able to access it now from other subnets, which is very good.
The Server Network is still inaccessible, but that's most likely because eth1 is on a different subnet and can't reach the default gateway, which is on the Management subnet. Does this all make sense/is there anything else that looks like I've missed something?
Thanks!