In case anyone else runs into a similar issue, I'm now up and running, and the database has indeed been migrated to the new v6.5.31.0 VM.
On restarting the new VM after the initial web configuration, and being given the "First time appliance configuration failed." error, I thought I'd log in to the Webmin web console for further investigation (enter the management mode of the VM, and choose "Enable/Disable Webmin interface" to toggle this on/off - by default it was disabled). Once in Webmin, there was an entry under Servers for ESET Remote Administrator, but this was blank, probably as to be expected. The Webmin Dashboard was flagging up that there were package updates available (290 of them), so I thought it was worth a try. I let these package updates download and install, and restarted the VM.
After this, I logged back on to the web console for VA configuration (hxxp://help.eset.com/era_deploy_va/65/en-US/index.html?config_va.htm), choosing "ESET Remote Administrator Server Appliance" again, for the initial set up. I only entered something in the password field, as shown below, leaving all the network setting fields blank (as I have done previously). "Submit" was clicked, the VM restarted, and the VM console then loaded without the "First time appliance configuration failed." error, showing "Server version: 6.5.417.0, Agent version: 6.5.417.0, Rogue Detection Sensor version: 1.0.1079.0). Huzzah!
On logging in to the new ERA web console (port 8443), the old database seems to be intact - computers and static groups are there, custom policies look to be copied across and in place, quarantine and threat logs seem to be there etc. The only thing I've seen so far that needed to be added, that wasn't present, was our license details under Admin -> License Management.
In conclusion, it looks like the package updates in Webmin saved the day. 8)