edit2: Technically your post as quoted is correct. However, in the context of the question, the implication (or inference) from the was that both would need to be convictions.
Yes, Andy, I quite agree that a FP could be taken and only the second offence dealt with in court where the "same occasion" argument can be made. It certainly has the advantage of ensuring one of the offences is sentenced by way of a FP. But I think that argument does not have much chance of success and I believe what little chance it has may be enhanced by presenting the two infringements to the Bench together - even though the overall financial penalty may be greater.
As you know, magistrates are, in the main, simple country folk who don't like to be baffled too much by complications!
Both give the option of accepting 3 points and a fine or going to court and neither say she has to go to court.
They may not say so but she
will be required to attend court at the very least for the single charge that will take her to twelve points. If the ticket office people do their job properly, if she accepts a fixed penalty for the second offence her acceptance will be rejected and court proceedings issued.