The Nottingham attack victims' families have accused the BBC of causing "trauma" and failing in a duty of care by making a documentary about the killings without asking them to contribute.
They said the broadcaster only informed them about the Panorama programme on 26 July.
The show is broadcast tonight and reports that a doctor had warned Valdo Calocane could "end up killing someone" three years before his rampage.
It also features the killer's mother and brother - who the BBC said have criticised the "broken" mental health system for a "tragedy that could have been prevented".
Calocane stabbed Barnaby Webber, Grace O'Malley-Kumar and Ian Coates in June 2023 and was convicted of manslaughter - rather than murder - due to his paranoid schizophrenia.
A statement from Emma Webber, on behalf of the families, said the BBC had failed to consider "the appalling trauma" the Panorama show would cause them.
It said it had given them little detail other than "big questions would be answered" and that Calocane's family would appear.
The BBC said in a statement that it has the "deepest sympathy" for the families and had contacted them with "an outline of its editorial focus".
It said the programme is "very much in the public interest" in looking at potential mental health failings and that it complied with editorial rules.
However, the families said they felt let down by the BBC and had "raised their concern at the lack of consideration at the appalling trauma this would cause them".
They said they did not want to stand in the way of investigative journalism but "believe the BBC have failed in their duty of care in any form of consideration for them".
"The matter was raised at the most senior level to the editor of the programme by both the families and also by the health secretary," their statement added.
The Panorama episode is being aired the day before a review is published into how health authorities dealt with Calocane in the years before his rampage.
'Valdo will end up killing someone'
He had been sectioned several times and the programme features medical notes from July 2020 warning he could kill.
It states: "Dr [redacted] observed that there seems to be no insight or remorse and that the danger is that this will happen again and perhaps Valdo will end up killing someone."
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Tap hereThe warning from a psychiatrist was noted down as a medical team reviewed his case during one of his stays in hospital.
The note is reportedly part of a summary of records his family received after he was sentenced.
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Calocane was detained in a high-security hospital - "very probably for the rest of his life" - in January after prosecutors accepted a manslaughter plea on the basis of diminished responsibility.
The victims' families criticised the decision, with Mr Webber's mother Emma telling Sky News in May: "We recognised he is very mentally unwell.
"However, as we've always said, we do not accept that he was not without ability to make rational decisions, and that he planned these heinous murders, and that he knew exactly what he was doing."
The Court of Appeal, however, ruled the sentence was not unduly lenient.
Police watchdog the IOPC is also looking into officers' previous actions in relation to Calocane before he committed the killings.
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