Speaking in NY, family spokesman Radd Seiger said: "We want to conduct an investigation into the FCO's decision to advise Northamptonshire Police that this lady had the benefit of diplomatic immunity".
She also emphasised that it was "not appropriate to meet her without therapists and mediators for us as a family but also for her".
'I genuinely do think he will look to resolve this in a way that will help us'.
UK Prime Minister Boris Johnson has demanded that Sacoolas should return to Britain, adding, "I do not think that it can be right to use the process of diplomatic immunity for this type of objective".
She said they "would still love to meet with her but it has to be on our terms and on United Kingdom soil".
'We believe this can only be achieved if Anne Sacoolas returns to England and engages properly with the justice system, where she will be treated fairly in a proper investigation of what happened to our son on that day, ' they said.
She expressed "that tiny little bit of hope" that Trump will "take another look and maybe work harder" for them despite the fact that during the Tuesday meeting, Trump did not suggest that he would send Sacoolas back to the UK.
In response to Ms Sacoolas's recent apology, Mr Seiger said: "I have seen those comments and obviously have no way of knowing whether those are indeed words which have been uttered by Ms Sacoolas herself".
"He's the one in control here, but we're the ones in control of our situation as much as we can be - we still want justice for Harry and we will take it as far as we possibly can to ensure that that's done".
A few days into their USA tour to raise awareness of their campaign, the family were invited to meet Trump at the White House.
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The US president has previously addressed the controversy, calling Mr Dunn's death a "terrible accident" and confirmed his administration would seek to speak to Mrs Sacoolas.
Harry's father, Tim, added that he did not feel that the meeting was a publicity stunt. We didn't get the answers that we wanted.
"He did ask two or three times", Dunn said.
"Our priority, as any parent will understand, is justice for our child".
"Friends tell each other the truth".
"The letter is worded very carefully, they're not saying it's just a change of heart - they are saying that it's an evolution", said Mr Seiger. "She waved down another vehicle", the spokesman noted.
Instead, Trump and national security adviser Robert O'Brien surprised the family by suggesting they meet with the woman in front of the White House press corps, said one of the couple's lawyers.
"None of us know how we were going to react, to have that put on us", Charles said.
Mr Seiger added: "If we're not satisfied, then we'll go to a judicial review and ask a High Court judge to review it all".
"(O'Brien has) heaped grief and pain on the family by making them go through this but not allowing them to get the closure they need by talking to Mrs Sacoolas before they can go onto the grieving stage", Stephens told BBC radio on Thursday.