I don't understand what the deal is with the North Gate and East Gate thing. Brockwell Park has about ten entrances and they'll all be open to the general public. Do we need to be checked in or something?
That link isn't completely accurate - Herne Hill, Tulse Hill and North Dulwich are Overground stations not Underground. Herne Hill is by far the closest station. Brixton tube station and then a few stops on the bus is also a good option.
For food there are too many good options in Brixton to list - but my top casual dining picks are Other Side Fried chicken burgers, Koi Ramen, Honest Burgers, Kaosarn (Thai) and Okan - the latter three are in Brixton Village which takes a bit if finding but is worth looking for. https://brixtonvillage.com/eat-drink/
In Herne Hill I'd recommend Sesami delicatessen and Blackbird Bakery for light bites, or Utter Waffle for something more substantial (normally packed out though). There's a very good market in Herne Hill on Sunday lunchtime with some good streetfood options - personal faves are the Ethiopian curry and the raclette - Moroccan-style wraps are great but sell out quickly. Artisan ice cream place in the station is good, but I personally can never be arsed with the huge queue.
There's a Sainsbury's Local just the other side of Herne Hill train station, on the road called Herne Hill, for water and snacks, and there's a biggish Sainsbury's at Brixton Water Lane very near the North entrance to the park.