I think C (Bruneval) company, which as you say was a a Patrol company was displaced by the Gurkha reinforcement company.The first report does raise questions - Unless things have changed, C Coy 2 PARA is not a rifle company. It is Patrols Pl, Sigs Pl etc. Sounds more like they were on the same tour, not part of the Bde (especially given the photo where none has a maroon beret).
There was a bit in Pegasus a while back about its reformation in 2 Para orbat as C (Bruneval) Company. As I am sure you had beaten into you at Browning barracks, C Company Second Parachute Battalion carried out the Bruneval raid, Operation Biting in February 1942. The Parachute Regiments first battle honour.