Fully agree with Box Hill, only trouble is often you can't get into the car park by Ryka's on a sunday as it is full of bikes.Possible Stepping stones at Westhumble.
Well to the west of the area I know but maybe our Littlehampton friends could confirm space- the Oyster Catcher pub on the Bognor road, just past Ford. Sunday morning be good one for this- Drop other halves at the Market and then down to the Pub after.

Smith and Western Horsham or the one on the top of Boxhill good but not for the faint hearted food wise- large portions.
The Round stone? between Littlehampton and Worthing- large carpark.
Dukes Head Beare Green - parking for about 25 cars max New people running it , will cater for Kids, dogs if on a lead and well behaved. A24 Dual Carriage way right outside
