Republican congresswoman Liz Cheney had a short answer for Donald Trump on Thursday, after the former US president endorsed a challenger for her seat in Wyoming.

“Here’s a sound bite for you,” Cheney wrote on Twitter. “Bring it.”

Cheney is a stringent conservative but is nonetheless one of two Republican members of the House committee investigating the 6 January assault on the US Capitol by Trump supporters seeking to overturn his election defeat. The other is Adam Kinzinger of Illinois.

Republican leaders withdrew cooperation after the Democratic speaker, Nancy Pelosi, rejected committee spots for Jim Jordan of Ohio and Jim Banks of Indiana, both potential witnesses given their closeness to Trump and role in advancing his lie that his defeat was the result of electoral fraud.

Cheney was also among a handful of Republicans in the House and Senate who voted to impeach Trump for inciting the deadly Capitol attack, making his second impeachment the most bipartisan in US history.

Cheney’s criticism of Trump has made her a Republican pariah, losing a leadership role, and her position has become a proxy war in the broader fight for control of the Republican party between Trump and his allies and more traditional figures – one that the Trumpist forces are largely winning.

In a statement on Thursday, Trump endorsed Harriet Hageman in the Republican primary for Cheney’s seat ahead of midterm elections next year. He also called Cheney “the Democrats [sic] number one provider of sound bites” and a “warmonger and disloyal Republican”.

Cheney is the daughter of former vice-president Dick Cheney, who promoted and presided over the invasions of Afghanistan and Iraq under George W Bush from 2001 to 2009. She has fiercely criticised Joe Biden for his handling of the withdrawal from Afghanistan last month.

Hageman, Trump said, “is a fourth-generation daughter of Wyoming, a very successful attorney, and has the support and respect of a truly great US senator, Wyoming’s own Cynthia Lummis”.

Lummis did not confirm her endorsement, but called Hageman “a fabulous choice for President Trump”.

Hageman said Cheney “betrayed Wyoming, she betrayed this country, and she betrayed me” when she set herself in opposition to Trump and Republican leaders.

Cheney has welcomed the prospect of a competitive primary, telling NBC in May that if Republican leaders wanted to “make the argument that the people of Wyoming should vote for someone who is loyal to Donald Trump over somebody who is loyal to the constitution, I welcome that debate”.

Hageman is not the only challenger to Cheney. Trump has also endorsed challengers to other Republicans who supported impeachment.