Is it proven that the Prophet (blessings and peace of Allah be upon him) used to bargain when buying and selling until he began to sweat?

Firstly: 

The Prophet (blessings and peace of Allah be upon him) was the best of people in attitude and the easiest to deal with, when he bought, when he sold, and when he asked for payment. Worldly matters were not his concern, blessings and peace of Allah be upon him, and neither were issues of buying and selling his main concern. But he used to walk in the marketplaces, and he used to buy and sell, teaching people in word and deed the etiquette of transactions and the characteristics that the trader should have of honesty, self-restraint, trustworthiness, good treatment of people and loving good for people. 

Secondly: 

The Prophet (blessings and peace of Allah be upon him) would bargain when buying – and bargaining means offering a lower price – not because he loved to compete for worldly gain, but he was the teacher from whom people learned the teachings of their religion with regard to business dealings and all other matters, so he used to explain to them that which concerned them in all their affairs, in both word and deed. 

Moreover, being careful about how one spends money and trying to save money in permissible ways is something to which people have a natural inclination, and there is nothing in that that is contrary to good manners or religious teachings. 

Reflect on the attitude of the Prophet (blessings and peace of Allah be upon him) and that which has been narrated from him in a saheeh report: 

Al-Bukhaari (1991) and Muslim (715) narrated from Jaabir ibn ‘Abdullah (may Allah be pleased with him) that he was riding on a camel of his that had become exhausted to the point that he wanted to let it go. He said: The Prophet (blessings and peace of Allah be upon him) caught up with me and offered supplication for me, and he hit it and the camel started moving (energetically) as I had never seen it move before. He said: “Sell it to me for one uqiyah.” I said: No. He said: “Sell it to me.” So I sold it to him for one uqiyah and stipulated that I be allowed to ride it home. When I arrived I brought the camel to him, and he gave me its price, then I left. He sent someone after me and said: “Did you think that I bargained with you in order to take away your camel? Take your camel and its price; they are both yours.”. 

It says in Subul as-Salaam: 

This indicates that there is nothing wrong with asking a man to sell you something that he has, or with bargaining. End quote.

 It was narrated that Suwayd ibn Qays said: Makhramah al-‘Abdi and I brought some garments from Hajar to Makkah. The Messenger of Allah (blessings and peace of Allah be upon him) came to us on foot and bargained with us for some trousers, and we sold them to him 

Narrated by at-Tirmidhi (1305) – he said: it is hasan saheeh. Also narrated by Abu Dawood (3336), an-Nasaa’i (4592) and Ibn Maajah (2220). 

Bargaining refers to haggling between the seller and the buyer when deciding the price, as it says in an-Nihaayah (2/425) 

This bargaining, in a moderate and reasonable manner, is a sign of maturity, sound mind and knowledge of the market price. This is something praiseworthy because such a person is less likely to be given an unfair deal or be deceived or cheated when buying and selling. 

Hence the fuqaha’ said: you may know that the son of a trader has reached maturity once he knows how to buy and sell and bargain. 

See: al-Mawsoo‘ah al-Fiqhiyyah (22/215) 

With regard to what is mentioned in the question about the Prophet (blessings and peace of Allah be upon him) bargaining until he began to sweat, and the like, we know no basis for this in any report from the Prophet (blessings and peace of Allah be upon him), and it does not fit his character and easy-going nature in buying and selling, or his kindness in all his affairs. 

And Allah knows best.