“Could you come with me” is perfectly correct. If not, then the BBC site I quoted is also wrong (it gives “Could you come with me?” as an example).
Google “could vs can” and see for example these online discussions: 1, 2, 3 ,4. Informally the difference is that “could” is ‘more polite’ and leaves the person answering the question more room to back out. Formally “could” is the subjunctive mood.
Perhaps this is an arbitrary “rule” invented by the setters of the PSLE. But if so, they are wrong. (Though of course, in Singapore, getting things right is less important than giving the teacher the answer she wants and getting your marks on an exam.)