The scandal-ridden New York Republican congressman George Santos has restated his determination not to be forced out of the US House, vowing: “I’m not leaving, I’m not hiding and I am NOT backing down.”

Santos broadcast his defiance on Twitter on Tuesday.

“I will continue to work for New York’s third district and no amount of Twitter trolling will stop me,” he wrote. “I’m looking forward to getting what needs to be done, DONE!”

Many Republicans in Santos’s district and state and in Congress, and most Democrats, would rather his career in the House was done, after an extraordinary run of sensational stories since his election in November.

Santos’s résumé has been shown to be largely made-up. Wild claims about his family history and business career have been debunked, apparent appearances as a drag queen in Brazil exposed. He has been accused of criminal behaviour, in part in relation to a charity nominally set up to help distressed animals. He has been accused of sexual harassment by a former aide.

Santos’s campaign finance filings are under investigation. This week, the New York Times reported that $365,000 was “missing”. The source of Santos’s personal wealth is also under investigation, as are past activities under a different name, Anthony Devolder.

On Tuesday, Noah Bookbinder, president of the watchdog group Citizens for Responsibility and Ethics in Washington, said: “Hundreds of thousands of dollars in totally unexplained spending of campaign funds seems like something that is likely to become a major problem for George Santos. We don’t know yet, but this kind of thing often ends up pointing to crimes.”

Under investigation at local, state, federal and international levels, Santos has admitted embellishing his résumé but denied wrongdoing and repeatedly said he will not resign.

Democrats have called for Santos to be expelled but expulsions from the House are extremely rare. Only five representatives have ever been expelled – three for fighting for the Confederates in the civil war.

Republican leaders accepted Santos’s withdrawal from two committee assignments, a step he said he took to concentrate on serving his district. But party leaders have not called for Santos to resign, taking cover behind ongoing investigations of his campaign finances.

In January, Santos supported Kevin McCarthy for House speaker through 15 votes forced by rightwing rebels. McCarthy now leads the House with a slim majority that would be in danger of reduction via any special election in New York.

Last week, a group of Santos’s Queens and Long Island constituents rallied in Washington, calling for his resignation.

The same day, at Joe Biden’s State of the Union address, Santos was confronted by Mitt Romney, the Utah senator and 2012 Republican presidential nominee.

Romney told reporters Santos was “a sick puppy” and said he told him he did not belong in Congress.

Santos said it was “not the first time in history that I’ve been told to shut up and go to the back of the room, especially by people who come from a privileged background”.

A week later, Santos remained defiant. But Ron Filipkowski, a former federal prosecutor turned political commentator, pointed to his likely electoral fate – and that of Republicans now seen to be allied with him.

“Even if he survives to next year,” Filipkowski wrote, “he will get destroyed in his primary. The only question now is how many of his [Republican] colleagues is he going to take down with him on his way out.

“Every Democrat running should be waiting for that pic[ture] of their opponent with a smiling George Santos.”