Whether or not Barry and the Llewellyn Davis family picnicked in Kensington Park (or any park) is moot. Assuming they did, Forster’s Finding Neverland, with a screenplay by David Magee, fictionalizes three picnics that mark significant moments in Barrie’s relationship with the Llewellyn Davises.
The first and most important picnic occurs in Kensington Park when Barrie walks Porthos and meets the Llewellyn Davies family (Sylvia and her boys George, Jack, Peter, and Michael). Though Sylvia Llewellyn Davies has packed a wicker basket and other gear, Forster and Magee treat the picnic just for show. For the narration, this is the precise moment when Barrie tries to engage Peter to believe in his imaginative world, making him think that Porthos is a dancing bear.
At the second picnic, Llewellyn Davies has settled under a great oak, watching Barrie help the boys to fly a kite. The object of his lesson is that the boys can do what they want if they believe hard enough. 
The third picnic is at a cricket match where the family has set a picnic on the grass. On the pitch, however, Barrie (dressed in cricket gear) is told by a friend that his relationship with Mrs. Llewellyn Davis is a scandal and that his play with the boys suggests pederasty. Barrie is undeterred, sexless. The screenplay is so cautious that no one seems to have any sexual urge except Mary Barrie, who leaves her husband for another man.
Food is inconsequential at all the picnics.
The cast: Johnny Depp as J.M. Barrie; Kate Winslet as Sylvia Llewellyn Davies, and the boys are Nick Roud as George; Joe Prospero as Jack); Freddie Highmore as Peter, and Luke Sipp as Michael. Porthos is St. Bernard.
See Marc Foster. Finding Neverland (2004). Screenplay by David Magee based on a story by Allan Knee; J.M. Barrie. The Little White Bird. London: Stoughton & Hodder, 1902; J.M. Barrie. Peter Pan or the Boy Who Wouldn’t Grow Up. London, 1904; Andrew Birkin. J. M. Barrie & the Lost Boys: The Real Story of Peter Pan. New Haven: Yale University Press, 1979.
*Henry Justice Ford. The Child’s Map of Kensington Gardens. In J.M. Barrie. The Little White Bird. London: Hodder and Stoughton, 1902. This fictional map is the Frontice for the first edition. Arthur Rackham’s Peter Pan’s Map of Kensington Gardens is a revised map of Peter Pan in Kensington Gardens from J.M. Barrie’s The Little White Bird (1910).