This assumes OP is willing to enter both private keys on the same system. That kinda defeats the purpose of each having a separate private key.
That can be done with two separate Bitcoin Core wallets with MultiSig descriptors containing: (
private_key1,public_key2) and (
public_key1,private_key2).
In that case, OP should sign the transaction with the first wallet, export the partially signed transaction and sign it with the second wallet.