Blocked on microsoft/vscode#43768 being finalised.
Update: The VS Code API works, however despite supporting multiple files, it is only ever fired once-per-file https://github.com/microsoft/vscode/blob/926fc23f139e4d1250dba494c6e825e4816e62f6/src/vs/workbench/api/common/extHostFileSystemEventService.ts#L148 concurrently. This results in multiple MOVE_FILE calls to the analysis server, which currently only supports one refactor at a time (a new refactor cancels the previous).
The implement this reasonably (and without causing a potentially-large number of sequential requests to the server) we need to either have the server support multiple files, or do more work on the server (eg. via LSP).