The phrase "while claiming that" is correct and usable in written English.
You can use it when introducing a quote or a claim from a source. For example, "John Smith, a noted historian, declared that the cause of the war was clear, while claiming that the other side didn't stand a chance of winning."
Rightwingers treated her as if she were actually planning to blight crops and hex her enemies, all the while claiming that they didn't believe in witchcraft.
He tweeted at Spelman incessantly, insulting her work and her intellect, all the while claiming that his initial attentions were just in the service of advancing Spelman's career.
"She frequently laughed, as she did often, while claiming that I had a mistress," he writes.
So she has been endorsing one unproven computer product while claiming that others do harm.
He (rightly, in my view) praises liberal democracy, while claiming that political revolution always ends in gulags and bloodshed.
Gov. James McGreevey has danced around the issue while claiming that he supports needle exchanges in principle.
"We ignore all these things at our peril," he says, while claiming that he's not seeing such a drain yet.
We'll declare victory and raise a second round while claiming that this had been our strategy all along".
Maciej, an ex-Poland international himself, condemned the Arsenal boss' recent treatment of his son, while claiming that he has made him a 'scapegoat'scapegoat
He was loth to condemn intolerance in neighbouring Iran, while claiming that Muslims in the West were not allowed to practise their religion freely.
While claiming that no policy proposals would be "prejudged," this Task Force spent most of its time on proposed restrictions on lawful firearms owners – honest, taxpaying, hardworking Americans.
