A while back many moons ago some people in our town raised the issue of the evidences for the hanafi method of offering the eid prayer and questioned its reliability resulting in some confusion in the minds of some people.
The very useful lecture notes to the hadith book Jami al-Tirmidhi by Mufti Taqi al-Uthmani came in very handy. I purchased the first 3 volumes a while back and found the notes very useful and informative. The 3 volumes of the first release (subsequent volumes have since been released, however I do not possess a copy of these) are helpfully annotated in Arabic with cross referencing by the Sheikh’s student who has done an excellent job, may Allah (Most High) reward him. Please find the section dealing with the number of takbirs in the eid prayer below.
The following extract can be found on (2/314) onwards of Dars al-Tirmidhi, only the evidences for the hanafi’s have been translated, and the Sheikh’s replies to the evidences of the other madhabs have not been included, some slight abridgements have been made.***
The Evidence of the Hanafi’s
The hanafi’s first proof is from the narration of Makhul in Sunan Abu Dawud that:
Abu Aishah who was the companion of Abu Hurayrah informed me that Saeed bin al-Aas asked Abu Musa al-Ashari and Hudhayfah bin Yaman regarding how the Messenger of Allah (Allah bless him and give him peace) would make takbir in the Adha and Fitr (prayers)? Abu Musa replied he would make takbir four times (similar) to the takbir in the funeral (prayer). Hudhayfah said: you are correct, Abu Musa said: I used to make takbir in this manner when I was amongst them…
Four takbirs are mentioned in this hadith, from these one is the opening takbir (takbir al-tahrimah) and three are extra. This hadith is equivalent to two hadith because it is mentioned that Hadrat Hudhayfah affirmed what Hadrat Abu Musa said.
Objection has been made to this hadith that it revolves around Abd al-Rahman bin Thawban who has been declared weak. The answer to this is that Abd al-Rahman bin Thawban is a narrator with regards to whom there is a difference of opinion, where as some of the scholars of hadith have declared him to be weak the same number of scholars of hadith have declared him trustworthy. Therefore Hadrat Duhaim and Abu Hatim have declared him trustworthy, and Imam Abu Dawud said with regards to him ‘there was good in him (kana fihi salamah) and he was someone whose prayers were answered’ and Ibn Maeen said : ‘there is no harm in him’ also Saleh Jazrah has declared him ‘truthful’ and Ibn Adiyy said: ‘despite his weakness his hadith should be recorded’ therefore his hadith are no less than the level of Hasan
The second objection upon this hadith is that the narrator Abu Aisha is unknown according to Ibn Hazm and Ibn Qattan. The answer to this is that he is the father of Muhammad bin Abi Aishah and Musa bin Abi Aishah. Hafidh Ibn Hajar has said with regards to him in Taqrib: ‘Abu Aishah al-Umawi, their mawla, the companion of Abu Hurayrah, acceptable (maqbul) from the second (level)’ also Hafidh said in Tahdhib regarding him : ‘Makhul and Khalid bin Madan narrate from him’ and it is agreed in the principles of hadith that from which person two narrators narrate, then his being unknown (majhul) no longer applies, therefore the objection is not valid, and the hadith is no less than the level of hasan
Imam Baihaqi  has made an objection that this hadith is actually a statement of Hadrat Ibn Masud, the detail of which is that this narration is narrated in Musannaf Abd al-Razzaq from Alqamah and Aswad bin Yazid that:
Ibn Masud was sitting with Hudhayfah and Abu Musa al-Ashari, Saeed ibn al-Aas asked them both regarding the Takbirs in the prayer on the day of al-Fitr and al-Adha, each of them said: ask this person, Hudhayfah said to him: ask this person meaning Abdullah Ibn Masud, so he asked him and Ibn Masud said: make takbir four times then recite then make takbir and bow, then stand up for the second rakat and recite, then make takbir four times after the recitation
From this it is known that this narration is a statement of Ibn Masud and is only narrated from him.
Allamah Nimawi has answered this by saying that reconciliation is possible between Abu Musa Ashari’s marfu narration and Hadrat Ibn Masud’s mawquf narration as Hadrat Abu Musa at the beginning was silent out of respect in front of Hadrat Ibn Masud, and when Hadrat Ibn Masud gave the legal ruling with regards to the issue then Hadrat Abu Musa in support of his statement gave his marfu narration. Then for arguments sake if it is accepted that this narration is only mawquf to Ibn Masud even then due to it not falling within the scope of analogy (qiyas) it will have the ruling of a marfu (narration). Then in this narration a group of the companions have agreed with Ibn Masud from which this narration gains extra strength.
The Hanafi’s second proof is the practice of Hadrat Ibn Abbas, Hadrat Mughirah bin Shubah, Hadrat Ibn Masud  may Allah be pleased with them and others 
Also a large number of the Tabieens position was in agreement with the hanafi’s 
The Hanafi’s third proof is the narration of Ibrahim Nakhai  who said:
The Messenger of Allah peace and blessings upon him passed away with the people differeing over the (number) of takbirs in the funeral (prayer)
He then further on said:
The difference continued until Abu Bakr passed away, and when Umar became leader and the difference of the people regarding this (issue) troubled him greatly, so he sent for the men from the companions of the Messenger of Allah peace and blessings be upon him and said: you are the companions of the Messenger of Allah peace and blessings be upon him, when you differ those after you will differ, and when you agree upon a matter the people will agree upon it…. So they discussed the issue among themselves and agreed to make the takbir in the funeral prayer like the takbir in the adha and fitr (prayers) consisting of four takbirs, and they agreed upon this.
So we learn that in Hadrat Umars time there was agreement (ijma) that there would be 4 takbirs in the eid prayer.
Allamah Ibn Rushd has written in Bidayah al-Mujtahid  that no there is no sahih marfu hadith regarding the number of takbirs in the eid prayer, therefore with regards to this he has copied the statement of Imam Ahmad bin Hanbal that: ‘there is no sahih hadith reported from the Prophet peace and blessings be upon him regarding the takbir in the eid prayer’. Ibn Rushd states that the different jurists (fuqaha) have based their opinions upon the practice of different companions. Also this difference is one of virtue, there is agreement that the prayer will be valid (with a different number of takbirs). And Allah knows best that which is correct
Dars al-Tirmidhi (2/314) onwards
 Sunan Abu Dawud (1/163)
 Uthman bin Saeed reports from Ibn Maeen that he said: weak, and Ahmad said: his hadith are reprehensible (manakir), al-Nasai said he: was not strong. Mizan al-Itidal (2/551)
 Maarif al-Sunan (4/439)
 Hafidh al-Zaylai in Nasb al-Rayah (2/214) after narrating this from Abu Dawud said: Abu Dawud was silent regarding it as was al-Mundhiri in his Mukhtasar, and Ahmad narrated it in his musnad
 In his Sunan al-Kubra (3/290)
 (3/290) it should be clear that this chain of narration does not contain Abd al-Rahman bin Thawban nor Abu Aishah
 al-Taliq al-Hasan ala Athar al-Sunan (p.254)
 Musannaf Ibn Abi Shaibah (2/174)
 Musannaf Abd al-Razzaq (3/295)
 Musannaf Abd al-Razzaq (3/293)
 Translators note: ten companions names are mentioned in the footnotes, those interested in the details are referred back to the original
 see Musannaf Ibn Abi Shaibah (2/172)
 Sharh Maani al-Athar (1/239)
 see Badhl al-Majhud (2/207)