Rishi Sunak has been accused of leaving Jagtar Singh Johal to "rot in jail" in India, after the Foreign Office ruled out intervening in the case.
Mr Johal's brother Gurpeet said the government lacked the "political will" to get him home.
The Dumbarton man is facing the death penalty, having been detained in India since 2017.
Mr Sunak is in Delhi for the G20 Summit but refused to commit to raising the case with Indian PM Narendra Modi.
And in a letter to Mr Johal's local MP, Martin Docherty-Hughes, Foreign Office minister Lord Ahmad said: "Having carefully considered the potential benefits and risks to Mr Johal of calling for his release, as well as the likely effectiveness of doing so, we do not believe this course of action would be in his best interests.
"With Mr Johal's trial under way, our assessment is that such a call would be considered interference in the Indian judicial process, which could jeopardise our ability to offer consular assistance in Mr Johal's case."
Lord Ahmad added that the UK government would continue to raise concerns with India and that British officials regularly visited Mr Johal to check on his welfare.
Foreign Secretary James Cleverly expressed similar sentiments in July when he wrote to Mr Johal's family, setting out why the UK would not be calling for his release.
MP for West Dunbartonshire Mr Docherty-Hughes said the prime minister had "chosen not only to prioritise a desperate bid for a free trade deal over the life of a British citizen, he has also chosen to snub a perfectly reasonable request to him made by more than 70 of his fellow MPs".
Earlier this week, a cross-party group of MPs urged Mr Sunak to intervene in the case during his trip to Delhi.
Mr Johal, a 36 year-old campaigner for Sikh human rights, travelled to India in October 2017 to get married.
While there MPs say that having arrested him "interrogators electrocuted him, and threatened to douse him in petrol and set him alight.
"To make the torture stop, Jagtar recorded video statements and signed blank pieces of paper."
These allegations have been denied by the Indian authorities.
The UN Working Group on Arbitrary Detention said he had been targeted because he had called for "accountability for alleged actions committed against Sikhs by the authorities".
He is currently facing eight charges of conspiracy to murder, linked to political violence in India.
Speaking to the BBC, Mr Johal's brother Gurpreet – a lawyer and Labour councillor – said Lord Ahmad was concerned but "clearly had no power to get things done".
"They say the case is a priority and they are looking for a resolution but fob me off with a junior minister.
"I'm fed up with being told there's nothing the government can do.
"My brother has been missing from this family for six years now… the prime minister and foreign secretary are leaving an innocent British national to rot in an Indian jail, because they lack the political will to do what it takes to get him home."
On the plane to Delhi, when asked if he would be raising the case, Mr Sunak said: "I'll be raising a range of things with Prime Minister Modi – this is something that, just so people are reassured, has already been raised on multiple levels on multiple occasions."
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