I think you need to add your custom info in the bitcore-node.json which is created in your node directory
{
  "network": "livenet",  // your coins network name
  "port": 3001,          // your coins port
  "services": [
    "bitcoind",          // required services
    "web"
  ],
  "servicesConfig": {
    "bitcoind": {
      "datadir": "/home/user/.bitcoin",         //your coins directory
      "exec": "/home/user/bitcoin/src/bitcoind" //your coins executable
    }
  }
}
So changing
"network": "livenet"
to whatever yours is called should do the job. Now you should be able to run your custom node with a different network.