Had a look at FR24 as I have noticed the odd Spitfire pop on there occasionally (BBMF do show occasionally - noticed one of their Hurricanes heading along the south coast to Lydd recently!), but nothing showed in the area between the time in question (12:20 GMT).
A likely candidate would be one of the Boultbee two seat Spitfires from Goodwood - I was at a cricket match at Arundel a few weeks ago and we enjoyed several mild interruptions by Merlin noises and an unmistakeable shape a mile or two away descending towards the circuit at Goodwood.