Even before Republicans took the House of Representatives, leading figures on the right of the party pointed to troubled waters ahead for Kevin McCarthy – or whoever else becomes the next House speaker.

Now Republicans have won their slim victory in the lower chamber of Congress, the next two years are likely to be chaotic. Controlling an unruly party with an extremely narrow majority will all but guarantee brutal tests every day, especially from the right wing.

Fighting among Republicans over who leads the House is already in full swing. On Tuesday, Clay Higgins of Louisiana, a member of the hard-right Freedom Caucus, would not tell Politico if he would back McCarthy.

But Higgins did say: “The speaker of the House, whomever he or she is, will be required to recognise the center of gravity of the conference itself. And the Freedom Caucus has moved that center of gravity to the right.”

Andy Biggs, a Freedom Caucus member from Arizona and an ardent backer of Donald Trump’s electoral fraud lie, challenged McCarthy, now minority leader, to be the Republican nominee for speaker.

“My bid to run for speaker is about changing the paradigm and the status quo,” Biggs tweeted, adding: “McCarthy does not have the votes needed to become the next speaker of the House and his speakership should not be a foregone conclusion.”

Biggs did not have the votes in secret leadership ballots on Tuesday, losing 188-31 to McCarthy. But the speaker’s role will not be decided till January and any candidate for speaker must attract 218 votes. That is a simple House majority, not confined to party lines, but Republicans are set to hold power by not much more and McCarthy must now win over his skeptics.

Backstage manoeuvers are in full swing. On Tuesday a moderate Democrat, Henry Cuellar of Texas, confirmed that consultants allied to McCarthy had been seeking his vote, via a switch to become a Republican.

The Republican party is in flux, after a predicted midterms “red wave” failed to materialize and with leaders under fire from all directions, each faction seeking to identify what went wrong and set course for the next two years.

Another prominent rightwinger, Matt Gaetz of Florida, is among those demanding aggressive action against Democrats, including investigations of Hunter Biden, the coronavirus response and immigration policy and swift impeachment of Joe Biden.

On Monday, Gaetz said of McCarthy: “I’m not voting for him tomorrow. I’m not voting for him on the floor. And I am certain that there is a critical mass of people who hold my precise view.”

On Tuesday, after McCarthy’s party vote victory, Gaetz told reporters the current minority leader “couldn’t get 218 votes, he couldn’t get 200 votes, he couldn’t get 190 votes today, so to believe that Kevin McCarthy is going to be speaker, you have to believe he’s going to get votes in the next six weeks that he couldn’t get in six years”.

Illustrating party ferment, however, an equally visible and extreme provocateur, Marjorie Taylor Greene of Georgia, has taken an opposing view.

Speaking to the former Trump adviser Steve Bannon, Greene said denying McCarthy the speaker’s gavel would be “bad strategy when we’re looking at having a very razor-thin majority”.

McCarthy has confirmed that if made speaker, he will restore to Greene committee assignments Democrats stripped over her extremist views and behaviour.

Greene also pointed to an outlandish idea, nonetheless circulating on Capitol Hill, that moderate Republicans could join with Democrats and install Liz Cheney, an anti-Trump conservative, as speaker.

Greene said: “We’ve already been through two years where we saw Republicans – Liz Cheney and Adam Kinzinger – cross over and join the Democrats and produce a January 6 committee.

“The danger is this: do we want to watch a challenge for speaker of the House simply because the ‘Never Kevin’ movement – just like we’ve seen a ‘Never Trump’ movement – do we want to see that challenge open the door to Nancy Pelosi handing the gavel to Liz Cheney?”

Cheney will soon leave the House, having lost her Wyoming primary to a Trump-backed challenger. But the speaker of the House does not have to be a member of Congress, hence a previous fringe idea that Republicans could put Trump in the role.

On Monday, Don Bacon, a Republican moderate from Nebraska, told NBC News “Cheney for speaker” was a non-starter. But he also said that if his party paralysed itself with partisan infighting, he would work with Democrats to install a moderate Republican speaker.

“I will support Kevin McCarthy,” he said. “But … I do want the country to work and we need to govern. We can’t sit neutral. We can’t have total gridlock for two years.”