Councillors rejected the request when they deliberated on the council's 10-year plan last week, but they did ask council staff to work with the trust and report back before the 2022-23 annual plan is adopted.
The asbestos roof was removed in 2017 as part of work to make the building safe, Dunedin City Council city services general manager Robert West said.
The council recognised the significance of the building to the community and would work with the trust to arrive at a suitable result, he said.
Smith said the trust was disappointed the funding request had not been accepted, but heartened that councillors Jim O'Malley, Steve Walker, Sophie Barker and Christine Garey had spoken in support of the council taking action.
"The point was well made by Cr O'Malley that whatever happens to the building, the council, as owner, has an obligation to deal with the asbestos and stabilise the bank behind, essentially making the site safe," Smith said.
"Our concern is the failure to see these factors as urgent, so that the trust can get on with re-roofing, leaves the raw structure exposed to the elements and amounts to demolition by neglect."
Smith said the trust was keen to work with the council and it was getting its own assessments.
"Our architect, Richard Naish, and an engineer are going through the building later this month so we can have a clearer understanding of the protection work needed."
The Sims building was constructed about 1880 and housed the Stevenson and Cook Engineering Company.
In recent times, it was leased by the city council to various community groups.
It has been unused since 2017.
The site is on reserve land, which must be kept for public use.
