Is it permissible for the father to register a house in the mother’s name in exchange for her share of the inheritance, on the condition that the mother be deprived of the inheritance upon death, given that the marriage contract is not officially recognized in Egypt and that the father has another wife?

It is your mother's right to inherit from her husband as long as the marriage contract is valid, even if it was not registered or recognized by the government. What your father wants by writing the house in her name is not valid if it is a will, meaning she would not own it until after his death, for this is an impermissible will unless the other heirs approve it after his death. However, if he intends to give it to her as a complete gift, such that she owns the house during his lifetime, then this is a valid gift if it occurs before his death illness. Regarding preferring one (child/wife) over the other in gifting, there is a difference of opinion among scholars.