Firstly: 

It is not permissible to give zakaah to Ahl al-Bayt. For more details of that please see the answers to questions no. 21981 and 11802. 

Secondly: 

If a person is given zakaah because he is poor and needy, then gives some of it as a gift to one of Ahl al-Bayt, there is nothing wrong with the latter accepting it because it has ceased to be zakaah and is now regarded as a gift. 

This is indicated by the report narrated by al-Bukhaari (5284) and Muslim (1075) from ‘Aa’ishah (may Allah be pleased with her) who said: The Prophet (blessings and peace of Allah be upon him) was given some meat and it was said: This is some of what was given in charity to Bareerah. He said: “It is charity for her and a gift for us.” 

It was narrated that Umm ‘Atiyyah (may Allah be pleased with her) said: A sheep was sent to Nusaybah al-Ansaariyyah and she sent it to ‘Aa’ishah (may Allah be pleased with her). The Prophet (blessings and peace of Allah be upon him) said: Do you have anything? I said: No, except what Nusaybah sent of this sheep. He said: Bring it, for it has reached its destination.

Narrated by al-Bukhaari (1446), Muslim (1073). 

Al-Bahooti said in Kashshaaf al-Qinaa‘ (2/294): The one to whom zakaah is forbidden may accept gifts acquired by those who are entitled to zakaah in the form as zakaah … End quote. 

Ibn Battaal said: … It is permissible for the poor person to dispose of charity by selling it, giving it as a gift and so on, because his ownership of it is valid. When Bareerah gave (something she had received as zakaah) to the household of the one who had manumitted her, namely ‘Aa’ishah, it became permissible for her [‘Aa’ishah] and for the Prophet (blessings and peace of Allah be upon him), and it changed from being charity to being the property of the one to whom it was given. Hence the Prophet (blessings and peace of Allah be upon him) said: “It is a gift from her to us and it has reached its destination” i.e., it became halaal (for them) when it passed from being charity to being a gift, because it is permissible to reward a gift by giving something like it and more, but that is not the case with charity. 

Al-Haafiz said in al-Fath: From this it may be understood that the prohibition is connected to the description, not to the thing itself. End quote. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Anyone who acquires wealth in a lawful manner has the right to give it to someone else in a lawful manner, even if it is not permissible for the latter to take it in the same manner as that in which the former took it. It was not permissible for the Prophet (blessings and peace of Allah be upon him) to accept charity, whether it was voluntary or obligatory (i.e., zakaah), but he ate from the charity that had been given to Bareerah, because Bareerah took it in a lawful manner, then the Prophet (blessings and peace of Allah be upon him) also took it in a permissible manner even though if he had taken it in the manner in which Bareerah had taken it, it would not have been permissible for him. This is a useful principle: If a person takes some wealth in a lawful manner then gives it to another person in a lawful manner, it is permissible for the latter even if, had the latter taken it in the same manner as the former, it would not be permissible for him. End quote from Sharh al-Kaafi. 

And Allah knows best.