Fair banding is supposed to be the reverse of selective - it should ensure that a school doesn't just have the top performers. That's why it is allowed for non-selective schools.
Using national scores for fair banding ensures that the intake is representative of the whole country, i.e. the whole of the United Kingdom. However, if children in Southwark are, on average, less intelligent than the rest of the country, using national scores will bias entry in favour of the more intelligent local children.
To illustrate, imagine the national average score is 100 but the average in Southwark is 90. If you have, say, 5 ability bands, using the national scores they may be 115+, 105-115, 95-105, 85-95, less than 85. Nationally there would be a roughly equal number of children in each band. However, a child getting the average score in Southwark would be in band 4, so over half the children in Southwark would be in bands 4 and 5. That means there would be much less competition for places in bands 1-3 making it easier for the more intelligent children to get places.
The reverse effect would also happen - if children in Southwark are, on average, more intelligent than the rest of the country there would be a lot of competition for places in bands 1 and 2, much less competition for places in bands 3-5.
Using the local scores ensures that the intake is representative of Southwark, so that there are roughly the same number of local children in each ability band.
In most areas I would expect little difference between the results using national scores and local scores. However, there may be a measurable difference in particularly deprived or particularly privileged areas.