As far as I know, there isn’t anything in InfluxDB that requires a connection to a public network except for telemetry reporting (“phone home”) which just reports information about your local instance back to InfluxData. To disable this functionality, just add the --reporting-disabled flag when you start influxd:
influxd --reporting-disabled
If this doesn’t work, this is definitely a head scratcher that will require some more digging. If this does fix it, it’s also something that should be reported as a bug.