Why not do both? See the show and afterwards you'll have plenty of time to go up to the stage door and meet the actors. Any specific show/actors you'd like to see/meet??
You will definitely have time to go from watching the show to SD. Actors do all sorts before leaving! Take mic's off, take make up off, get changed, organise travel, say goodbye to everyone, maybe even have a drink! :)
Bear in mind that the National has many exit options, not just the stage door and a main entry. I find it much harder to meet more famous actors there than other in other London theatres. Last three times I was there, I missed Ruth Wilson twice (she left through other doors) and nobody from the Angels in America cast came out either. I hope for you that Bryan will be different but this has been my NT experience so far. Instagram/Twitter is a good source to get realistic expectations on stage doors.