He interrupts the Eid takbeeraat to remind the people by quoting a hadith for which there is no basis

Firstly: 

We have previously explained the ruling on saying takbeer in unison before the Eid prayer, and that it is not prescribed. Please see the answer to question no. 127851 

Ibn al-Haaj (may Allah have mercy on him) said:

The established Sunnah is to say the takbeer loud enough that the individual and whoever is next to him can hear it. Doing anything more than that, to the extent of saying it in a very loud voice, is an innovation (bid‘ah), because nothing is narrated from the Prophet (blessings and peace of Allah be upon him) except what is mentioned above. Moreover, they say it in unison, and that is an innovation, because what is prescribed is only for each individual to say takbeer by himself and not in unison with anyone else. End quote. 

Al-Madkhal (2/285) 

Secondly: 

There is nothing wrong with one person saying the takbeer into the microphone, without the people following him and reciting in unison, for the reasons mentioned above. Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

If it is said to the people: We will delegate it to a specific individual, the mu’adhdhin or someone else, to recite the takbeer as prescribed over the loudspeaker without anyone following him and reciting with him in unison, I do not see anything wrong with that, because it comes under the heading of raising the voice with the takbeer and reciting it out loud, and it is a reminder to the heedless and the forgetful. It is well-known that if one of the people present says the takbeer in a loud voice, without a loudspeaker, no one will denounce him for that. Similarly, if he recites the takbeer over the loudspeaker, but without the people following him and reciting in unison, as if he is prompting them, and they wait for his takbeer so that they can say takbeer after him in unison, this is something for which there is no basis in the Sunnah. End quote. 

Majmoo‘ Fataawa wa Rasaa’il Ibn ‘Uthaymeen (13/987) 

Thirdly: 

There is nothing wrong with interrupting the takbeer and reminding the people to give zakat al-fitr before the prayer, so that anyone who has not given it may hasten to give it. The Prophet (blessings and peace of Allah be upon him) said: “Whoever gives it before the prayer, it is accepted zakaah, and whoever gives it after the prayer, it is merely ordinary charity.” Narrated by Abu Dawood (1609) and Ibn Maajah (1827); classed as hasan by al-Albaani in Saheeh Abi Dawood. 

With regard to quoting weak hadiths or those for which there is no basis, this is objectionable and is not permissible. Muslim narrated in the Introduction to his Saheeh (p 7) that al-Mugheerah ibn Shu‘bah and Samurah ibn Jundub (may Allah be pleased with them both) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever narrates a hadith from me knowing that it is false is one of the liars.” 

This is an emphatic warning against lying and having any involvement in lying, and indicates that whoever thinks that what he is narrating is most likely to be false yet still narrates it is a liar. End quote.

Telling lies about the Messenger of Allah (blessings and peace of Allah be upon him) is not like telling lies about anyone else. The Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever tells a lie about me deliberately, let him take his place in Hell.”

Narrated by al-Bukhaari (110) and Muslim (3). 

And Allah knows best. 

See also the answer to question no. 9464.