All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
You are not obliged to change your name as it does not include what necessitates changing it. Being named Fox is not forbidden in Islam, because being named with a name of an animal is only disliked in regard to the names of animals which are famous for having despicable qualities. Shaykh Bakr ibn Abdillah Abu Zayd said, “It is disliked to be named with the names of animals that are well-known for having despicable qualities, such as being named: Hanash (snake), Himaar (donkey), Kalb (dog)...”
Indeed, there was a Muslim scholar whose name was Tha’lab (Fox).
Also, it appears – and Allah knows best – that Christopher (Christ-bearer) is not an inadvisable name. If we assume that it is so, then a Muslim is not obliged to change his father's name, even if it includes a forbidden meaning in Islam, such as being enslaved to other than Allah.
Ibn al-Qayyim said in Tuhfatul-Mawdood, “The Companions used to be named with Bani Abd Shams (children of the slave of the Sun) and Bani Abd Ad-Dar (children of the slave of Ad-Daar [an idol), but the Prophet did not object to these names. Informing about someone’s name is broader than initiating a new name, so a name that already exists is permissible, whereas it is not permissible to initiate a new name [with a disliked or forbidden meaning].”
Allah knows best.