The Labour MP Richard Burgon hopes to relaunch the socialist Campaign Group as a leftwing rival to the Fabian Society, as Corbynites seek to regroup following Keir Starmer’s leadership victory.

Burgon, who is MP for Leeds East, came third in the deputy leadership race with 21.3% of the vote in the final round on Saturday, but his campaign, which won the backing of the Unite trade union, was widely regarded as a bid to become the spokesman for the left wing of the party from the backbenches.

“It’s a platform which he will extend and build on, to keep putting those ideas forward,” said an ally, who underlined the comparison with the Fabians.

Burgon’s most eye-catching policy was the idea that Labour members should be given a vote on whether the party’s leader should support any military action not backed by the UN. He called it a “peace pledge”.

Allies say he and John McDonnell, the former shadow chancellor, are working to beef up the longstanding Campaign Group, the caucus of socialist MPs behind Corbyn’s candidacy for the leadership in 2015.

The revamped organisation is expected to organise events and produce research and policy recommendations, as well as providing allied MPs with support. Burgon expects to play a leading role.

The Fabian Society, founded in 1884 to “educate, agitate, organise”, was involved in the creation of the Labour party and has remained affiliated to it, publishing research papers and acting as what it calls a “pluralist, non-factional forum” for developing political ideas.

Starmer’s shadow cabinet includes 15 Fabian members, according to the society.

Momentum, the grassroots campaign group Jon Lansman set up to support Corbyn’s leadership, is also under pressure to reinvent itself after failing to propel its chosen candidate, Rebecca Long-Bailey, into Labour’s top job.

Senior Momentum officials issued a statement on Tuesday saying the organisation could not continue as it had been, calling for “unity, reflection and comradely debate on the way forward”.

“At the present moment, our movement is in great peril of making the same mistakes we did after our defeat in the 1980s. We risk retreating from the challenge of making our ideas mainstream, fighting between ourselves and allowing our party to be taken over once more by centrists,” the statement said.

It appeared to be an attempt to pre-empt the launch the next day of a ginger group called Forward Momentum, which is calling for movement to be more grassroots-led and democratic, and less London-centric.

The group is expected to stand candidates for election to Momentum’s national governing council in May, in what appears to be an attempt to dilute Lansman’s control.

As well as failing to back the winner in the leadership race, Momentum’s candidates were defeated in parallel elections for two seats on Labour’s governing national executive committee (NEC).

The NEC has sweeping powers over party processes and policies. It took Corbyn more than two years to achieve a consistent majority on it.

A spokesperson for Momentum said: “We want to see a Labour government and we will use our campaigning tools to make that happen. We’ll continue to push big ideas. And we are going to work to ensure that Keir sticks to his promises.”

Starmer adopted much of Labour’s 2019 general election manifesto during his leadership campaign. He published 10 pledges, which included abolishing universal credit, scrapping tuition fees and increasing taxes on the top 5% of earners.

His shadow cabinet reshuffle, however, involved a comprehensive clearout of close Corbyn allies, with Burgon joining Jon Trickett, Dan Carden and Ian Lavery on the backbenches. Diane Abbott and John McDonnell had already made clear they would step aside.

Long-Bailey, who came second in the leadership contest, was one of the only survivors from the Campaign Group, and will now serve as shadow education secretary.

Many Labour veterans see the hand of the canny political strategist McDonnell behind manoeuvres now taking place to ensure leftwing voices remain prominent.

“John is mobilising on several fronts, and he’s adept at using proxies and ciphers,” said one senior party insider, joking that the shadow chancellor was “retiring to spend more time on politicking”.

Another suggested McDonnell and Burgon were coordinating their efforts, and would take a “comradely” approach to Starmer.