I don't think Neptune is supporting websocket. Could the "neo4j browser" use standard https request based communication?
As I was trying to do ":server connect", passing bolt+s://name.cluster-xxx.region.neptune.amazonaws.com:8182 and I got an error message: "Browser will attempt to open a websocket connection to .."
Amazon Neptune has it's own notebook based utility for interacting with the database called the Neptune Workbench. Workbench can be used either within a Neptune Notebook instance (preferred method - https://docs.aws.amazon.com/neptune/latest/userguide/graph-notebooks.html) or via the deployment of the Graph Notebook project (https://github.com/aws/graph-notebook).