Why did Galileo decide to flame Fr. Grassi over the latter's theory of comets, making sure to lose the Jesuits’ support?

The short, simple answer is: personal pride and professional disagreement.

BACKGROUND

In the latter half of 1618, three comets appeared

Jesuit mathematicians at the Roman College studied the comets with great care. They gathered information about their appearance from other Jesuits across Europe and held public meetings in Rome to announce their findings.

Source: James MacLachlan, Galileo Galilei: First Physicist

Next,

...in 1619 a pamphlet entitled An Astronomical Discussion of the Three Comets of 1618 appeared. The work was officially anonymous, but it was well known that the author was the man who now held the chair at the Collegio Romano...., the Jesuit priest, Orazio Grassi.

Source: Atle Naess, Galileo Galilei: When the World Stood Still

Galileo had not observed any of the comets as he was in poor health at the time. It was only after Grassi's work was published that he was stirred into action.

GALILEO'S FIRST RESPONSE

Discourse on the Comets, Galileo's response, was published using the name of one of his students, Mario Guiducci, though the work was clearly Galileo's. Why this pamphlet was a fierce attack on Grassi's work is discussed by both MacLachlan and Naess.

First, the astronomical side. One key issue was (briefly), as stated by Naess,

in the discussion that raged in Rome’s scientific and ecclesiastical circles, Grassi’s assumptions were portrayed as a weighty argument against Copernicus. This irritated Galileo...

Galileo followed the position of Copernicus, considered heretical by the Roman Catholic Church, and was in fact put under house arrest in 1633 for the remainder of his life. Other disagreements centered around telescopic observations (on which more below).

Second, Galileo felt his pride and position were at stake.

Galileo and his friends decided that they could not let the Jesuits take over the leading role in astronomy. Until then, Galileo had clearly been the paramount astronomical observer in Europe.

Source: James MacLachlan, Galileo Galilei: First Physicist

This is supported by Naess, who says Galileo

felt his position as Europe’s most fashionable astronomer under threat. He was getting enquiries about the comets from several quarters including the French court. And he had nothing to say about them –whilst Grassi was bringing out new observations and theories.

Galileo also felt slighted that Grassi did not once mention him in his pamphlet.

Naess concludes that

..the main motivation for Galileo’s contention was probably psychological rather than astronomical. Grassi had promulgated an argument based on telescopic observations, namely that comets were not much enlarged by the instrument and therefore had to be correspondingly further away, an argument he believed not everyone had understood and accepted.

With his highly developed sense of pride Galileo managed to construe this as an attack upon himself – on the maestro of the telescope, the first, most skilful and practised telescope-user of them all!

GRASSI HITS BACK

In fact, the pamphlets of both Grassi and Galileo contained fundamental errors and the 'healthy' exchange of views did not end there. Deeply offended by Galileo's pamphlet, Grassi responded later the same year (1619) as MacLachlan relates:

Grassi became thoroughly annoyed with Galileo....So Grassi quickly published a 70-page attack on the ideas of Galileo that Guiducci had reported. Grassi's book was called The Astronomical Balance: Weighing Galileo's Opinions on Comets. Grassi did not identify himself as the author. He attributed it to a fictitious pupil of his, whom he called Lotario Sarsi. Grassi went through the Guiducci report point by point, raising every objection he could think of.

GALILEO'S THE ASSAYER

It was some time before Galileo responded but when he did, he produced a major work.

Galileo's Roman friends, especially in the Lincean Academy, urged Galileo to reply in kind. He delayed for several years. Finally, in October 1622, three years after the controversy had begun, Galileo sent his completed manuscript to the Linceans for them to publish in Rome.

This book, the Assayer, was finally published in October 1623 and "is generally considered to be one of the pioneering works of the scientific method"