Keir Starmer has seen off a major Labour rebellion over a bid to force a parliamentary investigation into his appointment of Peter Mandelson, but many of his own MPs warned he was running out of political capital.

After Downing Street deployed its full weight to force Labour MPs to block a referral to the privileges committee over the scandal, some angrily accused Starmer of leaving them facing accusations of a “cover-up”.

Previously loyal MPs warned the prime minister to tread carefully, particularly after what are expected to be a damaging set of election results for Labour next week.

“He’s in the last-chance saloon and the last few days haven’t improved his prospects of survival,” one minister said, while another added: “Keir only has so much credit in the bank with the backbenches now, so he needs to spend it wisely.”

On a day of high jeopardy on Tuesday, Starmer’s former chief of staff, Morgan McSweeney, and the Foreign Office’s former permanent secretary, Sir Philip Barton, prompted yet more questions over how much pressure had been put on officials to accelerate Mandelson’s posting to Washington.

While the attention of Labour MPs will turn to limiting the fallout at the elections, the scandal is likely to raise its head once more next month, after the intelligence and security committee (ISC) announced it had finished reviewing key government documents.

But after days of intense pressure, Downing Street was taking temporary relief from the failure of a vote – tabled by the Conservative leader, Kemi Badenoch – on whether the privileges committee should consider whether Starmer had misled the Commons over Mandelson’s appointment.

The government won the vote by 335 votes to 223, a majority of 112, with insiders feeling some relief that senior figures such as Angela Rayner opted to keep their powder dry.

However, 15 Labour backbenchers supported the motion, mainly from the left of the party and with track records as rebels, and there was concern that up to 53 MPs did not vote, though not all of these will have been abstentions.

One of the rebels, Emma Lewell, the MP for South Shields, criticised the decision to whip Labour MPs to block the motion. “It has played into the terrible narrative that there is something to hide and good, decent colleagues will be accused of being complicit in a cover-up,” she said.

In a damaging revelation for the prime minister, it emerged on Tuesday that Christian Turner, the UK’s new ambassador to the US who took over from Mandelson, had described Starmer as having been “on the ropes” over the scandal.

He told a group of students in private remarks in February that Starmer’s future had looked “quite touch and go” but that he was a “stubborn guy” who would be unlikely to quit of his own accord.

“The moment I would look to is the May elections,” the Financial Times reported Turner as saying. “If Labour does very badly … I suspect the party will be able to go over that threshold and remove him – seems to me to be the conventional thinking.”

And on another day of damaging disclosures to the foreign affairs select committee on Mandelson’s security vetting, McSweeney admitted that Foreign Office officials came under intense pressure to expedite the posting, but denied they were forced to “skip steps” in security vetting.

Starmer’s former chief of staff, who resigned earlier this year over the scandal, acknowledged he had asked Barton, then the top official at the department, to conduct the process “at pace” but not to do anything “improper”.

In a rare appearance before MPs, McSweeney said: “There is a real difference between asking people to act at pace and asking people to lower standards. We never did that. We never asked people to skip steps at any part of the process … it was all about, can we do this at pace, not, can we do anything improper.”

McSweeney told MPs that learning the extent of Mandelson’s ongoing links with Jeffrey Epstein – after he had been questioned about red flags raised by the due diligence process and sent to Washington – was like a “knife through my soul”.

The former senior aide said Starmer would not have gone ahead with the appointment had he known the full truth. While he acknowledged that revoking Mandelson’s posting over his failure to get security clearance would have been “embarrassing” for the government, it would have been “far preferable” to allowing it to proceed.

Many Labour MPs are angry that Downing Street, which had been aware at the time that Mandelson was close enough friends with Epstein to stay overnight at his house, decided to send him to Washington regardless.

McSweeney admitted he had made a “serious error of judgment” in advising Starmer to appoint the former Labour minister, but said he had felt his “experience, relationships and political skills”, including on trade, could serve UK interests as Donald Trump re-entered the White House.

In his own evidence to the committee, Barton said No 10 had seemed “uninterested” in the vetting process around the appointment, and there were no avenues for him to express his concerns.

Asked if he was under pressure to get the vetting done quickly, he said: “Absolutely … I don’t think anyone could have been in any doubt in the department working on this that there was pressure to get everything done as quickly as possible.”

He denied having received any phone call from McSweeney – long rumoured – which had asked him to “just fucking approve it”. McSweeney told MPs that such Westminster rumours were “corrosive” to faith in the political system.

Amid speculation that Starmer is considering a reshuffle after the local elections on 7 May, the prime minister is said to have told Rayner earlier this month that he hoped she would rejoin his cabinet, according to he Daily Telegraph.

The Guardian understands that no offer of returning to the cabinet has either been made or accepted.

Starmer has previously said he would like Rayner to return to the cabinet, calling his former deputy “hugely talented” and “the best social mobility story this country has ever seen”.

Rayner resigned as deputy prime minister and housing secretary in September after Starmer’s ethics adviser, Sir Laurie Magnus, found she had breached the ministerial code over her underpayment of stamp duty on a flat in Hove, Sussex.