One of the less dignified spats in US politics has rumbled onwards as the Wyoming Republican Liz Cheney responded to a bizarre insult from Donald Trump.

“I like Republican presidents who win re-election,” Cheney tweeted on Sunday, with a picture of George W Bush.

Bush beat John Kerry for re-election in 2004. Cheney’s father, Dick Cheney, was vice-president to Bush.

Liz Cheney’s tweet was a response to an image released by Trump on Thursday. Under the heading “ICYMI: Must-See Photo”, a Trump-affiliated political action committee sent out a Photoshopped image which spliced Liz Cheney and George W Bush.

Trump could not tweet it himself, as he remains barred from the platform for inciting the deadly assault on the US Capitol on 6 January. Cheney voted to impeach Trump over his role.

In an interview with CBS’s 60 Minutes later on Sunday, Liz Cheney previewed her re-election campaign in 2022, as forces aligned with Trump try to unseat her.

“I think it’s going to be the most important House race in the country in 2022,” she said. “And it will be one where people do have the opportunity to say, We want to stand for the constitution,’” Cheney said.

“A vote against me in this race, a vote for whomever Donald Trump has endorsed, is a vote for somebody who’s willing to perpetuate the big lie, somebody who’s willing to put allegiance to Trump above allegiance to the constitution, absolutely.”

In a surprise admission, Cheney also said she had been wrong about gay marriage, which she opposed ahead of a Senate race in 2013. Her objections caused a rift with her sister, Mary, whose spouse, Heather Poe, said Cheney’s position was offensive, posting on Facebook: “I always thought freedom meant freedom for EVERYONE.”

Cheney told CBS: “I was wrong. I was wrong. It’s a very personal issue – and very personal for my family. And my sister and I have had that conversation … Freedom means freedom for everybody.”

While still opposed to gun control, abortion and the Affordable Care Act, and repeating to CBS her opinion that waterboarding is not torture, Cheney finds herself allied with Democrats over the Capitol riot.

Trump was then 14 days away from ceding the Oval Office to Joe Biden, who denied him a second term with a convincing election win. Trump did not and has not conceded defeat.

At a rally near the White House on 6 January, Trump repeated lies about electoral fraud and told supporters to “fight like hell” to overturn the result. Five people died in and around the ensuing invasion of Congress. More than 650 have been charged.

Cheney was one of 10 House Republicans to vote for Trump’s impeachment – his second, unique among presidents – but because only seven Senate Republicans voted to convict, Trump remains free to seek re-election.

Only one president – Grover Cleveland, a 19th-century Democrat – has won the White House again after losing a re-election bid. Trump, however, remains the dominant force in the Republican party, staging rallies, repeating lies about electoral fraud, handing out endorsements, attacking enemies within the party and without and presiding over a growing campaign war chest.

The spliced Bush-Cheney image was accompanied with a link for donations. It was also displayed at a rally in Georgia on Saturday.

Cheney’s response was lauded on Twitter by Meghan McCain, another daughter of a Republican establishment figure attacked by Trump, the late Arizona senator and presidential candidate John McCain.

With Adam Kinzinger of Illinois, Cheney is one of two Republican members of the House select committee investigating the 6 January assault. On Thursday, it announced subpoenas for figures close to Trump including former chief of staff Mark Meadows and former White House strategist Steve Bannon.

Cheney is among Republicans opposed to Trump to have attracted primary challengers. Earlier this month she responded to Trump’s endorsement of Harriet Hageman, a Wyoming attorney and Cheney family friend, with another pithy tweet.

“Here’s a sound bite for you,” she wrote. “Bring it.”

On CBS, host Lesley Stahl asked if Cheney had received private support from members of Congress.

“Yes,” Cheney said. “Both in the House and the Senate.”

Asked about her continued membership of a party in which 78% do not think Biden won the election, and in which vaccination misinformation and conspiracy theories are rife, she said that was why she was speaking up.

“When you look at the spread of these mistruths and the spread of the disinformation, you know, silence enables it. Silence enables the liar. And silence helps it to spread.

“So the first thing you have to do is say, ‘No. I’m not going to accept that we’re gonna live in a post-truth world.’ It’s a toxin in our political bloodstream … and it’s a really serious and dangerous moment.”