Charlotte Charles said her and the rest of Harry's family just want agencies involved in the case to "tell us the truth" and "learn that this cannot happen again".
The family's spokesman Radd Seiger told the PA news agency their meeting at the White House on October 15 ended with the President telling him that Secretary of the Treasury Steven Mnuchin was "all lined up to get the cheque ready".
Mr Seiger said the more he thought about the President's words, "the more shocking it is".
Harry, 19, was killed when his motorbike crashed into a car outside RAF Croughton in Northamptonshire on August 27.
The suspect in the case, 42-year-old Anne Sacoolas, left the country after the crash and claimed diplomatic immunity - which the family's lawyers dispute.
On Thursday, Mrs Charles and Harry's father Tim Dunn met Northamptonshire Police and Crime Commissioner (PCC) Stephen Mold to discuss the conduct of the force's Chief Constable Nick Adderley throughout the investigation into their son's death.
After the meeting Mr Mold admitted there were lessons to be learned from the investigation and said he would look at ways in which the family could be more supported.
Speaking to PA after Mr Mold's visit, Mrs Charles said: "It's harder to lie and withhold information and it's easier to tell the truth.
"That's all we want. We want her (Mrs Sacoolas) to come back and we want the systems all the way around it to just tell us the truth."
Asked if a cheque from Mr Trump would help them, Mrs Charles said: "No it's not. It's not going to bring Harry back.
"Justice has to be done. They all need to learn that this can't happen again and they need to learn that things need to be put into place to stop this.
"No-one should suffer over and above the loss of a loved one. It's enough to cope with that but it's absolutely the last thing we want.
"We're doing it because we'd be letting other people down if we didn't and we'd be letting Harry down if we didn't."
Asked if their position had changed on how they wanted the case resolved, Harry's father told PA: "Of all the things that are going on, for me - she still comes back to the UK and faces the justice system.
"It's frustrating. It is frustrating - they must see there's something not right. Why are they making it difficult?
"Why are they not just coming out with the information with how this has happened?
"I don't understand why they are making us fight for this information - it doesn't make any sense to me at all."
Despite the family's objectives remaining the same, Mr Seiger told PA their legal fight was only just beginning - saying it was clear that the US administration were "expecting to pay the bill".
Speaking about the family's trip to the White House last month, Mr Seiger said: "This is effectively what Trump said to me in the White House - he said Steven Mnuchin is standing by ready to write a cheque.
"So clearly the government are expecting to pay the bill.
"It was almost as if he let it slip out. When he said 'we've got the driver here', he basically meant we're all going to have a big hug and a kiss and I'll get my Treasury guy to write a cheque.
"That's how it was. On the day it just didn't register with me but the more I think about those words, the more shocking it is.
"In not so many words, he turned to Robert O'Brien and he said something like 'the Treasury are all lined up to get the cheque ready aren't they Robert?' - or words to that effect."
Reporting by PA.