Configure bitcored to work with regtest

Correct me if I am wrong, it seems bitcore only supports livenet (mainnet) and testnet (testnet3) network.

Is there a way to configure bitcored and bitcore-wallet-service to work with regtest?

Yes, you can use regtest, there are a suite of tests that use regtest: https://github.com/bitpay/bitcore-node/tree/master/regtest