The judge leading a High Court hearing over the 11-month-old's future today lifted an order which had previously barred United Kingdom media from identifying doctors involved in the case.
But the couple say there is new evidence and want Mr Justice Francis to carry out a fresh analysis of their case.
They have also failed to persuade European Court of Human Rights judges to intervene.
"It will be a horrific affront to every parent's innate right to care for their children if British courts refuse to allow Charlie Gard's parents to pursue offered treatment for their son", said King. "Congress should pass my resolution and take a firm stand for the rights of parents and for the life of little Charlie".
In April, the judge ruled in favour of Great Ormond Street and said Charlie should be allowed to die with dignity. Two hours into the High Court hearing, questions the judge prompted tensions to boil over.
He is due to stage further hearings later this month following next week's gathering of specialists.
Appearing via video link from NY, the U.S. doctor offering the treatment, who can not be named for legal reasons, said new data and analysis of his research work since the case was first heard in April had given a better perspective on the chances of the treatment's success.
"This case is about Charlie's rights and what the evidence tells us that they require", he said.
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His wife was later released but Turkish authorities charged him with "membership in an armed terrorist organization". None of that mattered during the three-and-a-half months she was in jail.
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The order bars Kardashian from contacting Chyna or coming within 100 yards of her home, vehicle or workplace. When asked why she sent the reclusive star the footage: "I've been broken up with Rob since December".
Before the start of the latest hearing, the family of Charlie condemned as "abhorrent" any threats of violence that might have been made against doctors or judges.
Opening the proceedings on Friday afternoon, Mr Justice Francis thanked the family for the appeal for calm. Additionally, Hirano claimed his treatment could improve brain function a "small but significant" amount.
A team of seven worldwide medical experts had alerted the hospital that fresh, unpublished data suggested that an experimental drug could improve Charlie's brain condition. Charlie's parents had initially sought to take their son to the United States for the experimental treatment.
"Ten days ago, virtually no one thought that Charlie Gard would live another day", Rev. Patrick Mahoney said in a Thursday Facebook live video.
Lawyers representing Charlie's parents objected to the meeting being staged in the absence of both parents.
Michio Hirano, MD, a neurologist at Columbia University Medical Center, will travel to London to examine the 11-month-old Charlie, who has mitochondrial DNA depletion syndrome, a progressive and fatal disease that causes muscle weakness and loss of motor skills.
Speaking on behalf of Charlie's family, spokesman Alasdair Seton-Marsden said outside court: "Today GOSH tried to block the parents of Charlie Gard from attending a meeting about their own child". Her parents, who wanted her kept alive, and her husband, who wanted to remove her feeding tube, were in a high-profile legal battle that ended when the courts sided with her husband. However, the hospital later agreed to consider new evidence involving Charlie's condition before taking him off the ventilator.