Is it permissible to pledge that the purpose of a visit request is for a family visit and not for Hajj or Umrah, while intending to perform Hajj, and would Hajj in this case be considered a complete Hajj?

The issue of regulating entry into a country falls under restricting what is permissible for the public interest. If you make a commitment to something, the basic principle is to fulfill it, due to the Prophet's (PBUH) saying: "Muslims are bound by their conditions." You can bring your wife for a family visit and then try to obtain a permit for Hajj, seeking Allah's help and supplication. If the regulations prevent you, then your wife is considered not able and Hajj is not obligatory for her. As for whoever manages to perform Hajj in violation of the regulations and completes the obligation fully without harm, we hope for him a complete reward, and his violation of the regulations does not diminish the reward of his Hajj.