The fire alarm was set off more than once, disrupting the talk, until security staff advised they move to a higher floor.
As she left after the event, Prof Sullivan said she heard “howls of ‘shame on you, shame on you'”.
The police told the BBC they had attended the event and said no arrests had been made.
The letter that has now been sent by Prof Sullivan’s lawyers says the University of Bristol had 15 months to find a secure location on campus for the talk.
It adds that documents disclosed by the university reveal there had been complaints made by its LGBTQ+ staff network against her speaking.
Prof Sullivan says she has raised many of her concerns with the Office for Students, which as regulator of universities in England has the power to fine institutions if they fail to uphold freedom of speech.
Earlier this year, the OfS handed out a record fine of £585,000 to the University of Sussex, warning that its transgender and non-binary inclusion policy had a “chilling effect” on freedom of speech.
Kathleen Stock had previously left her job as professor of philosophy at Sussex following protests against her gender-critical views, which generally say that a person’s sex cannot be changed and takes precedence over their gender identity.
Following the Sussex fine, the University of Bristol withdrew a similar policy which said all staff were responsible for removing “transphobic and anti-trans material” from campus.
In August, a new freedom of speech law came into force, strengthening the legal obligations on universities to uphold freedom of speech.
The government has not yet brought in the complaints process promised as part of the stronger powers.
Prof Sullivan has told Arif Ahmed, the OfS’s director of free speech, she is concerned that “gender ideology” led to Bristol’s “actions to obstruct and frustrate my talk”.
She told the BBC that senior managers at the university “could have apologised, and they could have said we intend to do better in future”.
“That is what I want from them,” she said.
A University of Bristol spokesperson said the talk went ahead safely and in line with its “strong commitment to upholding free speech”.
“Although protesters caused unacceptable disruption, appropriate measures were in place to enable the event to continue and to protect the speaker and attendees,” they said.
The university added that the vice-chancellor had subsequently met with Prof Sullivan.
“We refute claims that we failed to protect her freedom of speech; every action we took was in support of this and the restrictions she outlines were all necessary for public safety,” the spokesperson said.
“Our stance is clear – free speech must be lawful. There is no protection within the law for abusive speech that incites violence, harassment or discrimination.
“Clearly the intimidating behaviour of protesters was not peaceful and we have condemned their unacceptable behaviour.
“We will take disciplinary action if anyone from the university community is identified as being involved.”