No.
As has been pointed out, "No Insurance" is an "absolute" offence. You either have it or you do not. If you do not you are guilty. It is absolutely the driver's responsibility to ensure that cover is in place. Simply being told that it is is not sufficient. The driver must ask to see the documentation before driving.
Because of this "Special Reasons" not to award points or to disqualify are rarely, if ever, successful for the offence and they would certainly not be successful in the circumstances you describe. Your daughter can give it a try, but she will have to decline the fixed penalty offer (if it is made) and go to court. She will then be liable to a penalty as I have described in my earlier answer and it is most unlikely that her Special Reasons application will succeed.
One other avenue to try to avoid falling foul of the “New Drivers” revocation process would be to ask the Bench to impose a short disqualification (14 to 28 days would be appropriate) instead of points. Oddly, the regulations do not apply to drivers who have been disqualified but only to those who accumulate six or more points. Disqualification is within the magistrates’ powers, but they would be sentencing outside their guidelines and they may not be too keen. I have seen it succeed, but your daughter would be well advised to employ a solicitor to put her case. There is, of course, still no guarantee of success.