For the aircraft I fly, in identical conditions, calm wind, 25deg C and QNH1013…. Southampton RWY20 is limited by obstacles in the climb out. Southampton 02 and Southend 23 and 05 not so.
Runway length is often not a limiting factor in aircraft performance. Performance restrictions will be the most limiting of runway length, width, slope, surface, and obstacles.
The main benefit to the Southampton extension will be to put the obstacles further from the start of the 20 takeoff run, making them less likely to be limiting.
Link to CAA obstacle chart for Southampton;
NATS EGHI Obstacles