Your counsellors are mistaking correlation for causation. Cambridge generally looks for three A levels, but, if applying to read Maths, Physics, Engineering, or Computer Science, taking Further Maths as a fourth A level may well be useful.
Taking the A levels in one go would probably be the better choice.
Cambridge does not discriminate against applicants from independent schools, despite false claims to the contrary by some media orgs. Private school pupils still obtain a disproportionate number of the places available at Cambridge, Oxford, and some other competitive universities.