I think trying to comb the idea of the surreal apart from the art movement is an error in this case. HCB was surely sophisticated enough to not be a slavish devotee of a specific group of people, but was surely open minded and creative enough to be influenced by their take on an underlyng ideas and to make something new of them (both the ideas and the influences).
Photography, especially as pursued by HCB, is a radically different medium from, say, painting. The methods of the surrealist painters would have been almost useless to him.
ETA: That said, when I say "HCB was a surrealist" I mean rather more than "HCB was a photographer and photographers are surrealists"
I mean that HCB enjoyed a good visual joke, a peculiar juxtaposition of objects in the frame, random-seeming echoed shapes and lines. This leans more in the direction of "surrealist art movement" than in the direction of "photos are 'beyond-real'".