Is it permissible to marry a second wife secretly without officially registering the contract, provided there are witnesses and a guardian for the woman? And if the woman is a Muslim of Western origin whose father is a Christian who rejects Islam, who then is her guardian?

It is permissible for you to marry a second wife without registering the contract with the civil status authorities, as polygyny is permissible with the Sharia-compliant controls. As for the Muslim woman you wish to marry, her disbelieving father has no guardianship over her. Rather, her guardianship transfers to the closest Muslim guardian. If none is found, a Sharia judge should marry her off.