workspace-mcp: strip /workspace prefix for API GW proxy route

src/lambdas/workspace-mcp/handler.py

@@ -111,4 +111,23 @@ class _ActorCredentialsMiddleware:
         await self._app(scope, receive, send)
 
 
-handler = Mangum(_ActorCredentialsMiddleware(_app), lifespan='auto')
+class _PathPrefixStripMiddleware:
+    """Strip /workspace prefix so API GW proxy route maps to FastMCP's /mcp path."""
+    _PREFIX = b'/workspace'
+
+    def __init__(self, app):
+        self._app = app
+
+    async def __call__(self, scope, receive, send):
+        if scope['type'] in ('http', 'websocket'):
+            path: str = scope.get('path', '')
+            if path.startswith('/workspace'):
+                scope = dict(scope)
+                scope['path'] = path[len('/workspace'):] or '/'
+                raw: bytes = scope.get('raw_path') or b''
+                if raw.startswith(self._PREFIX):
+                    scope['raw_path'] = raw[len(self._PREFIX):] or b'/'
+        await self._app(scope, receive, send)
+
+
+handler = Mangum(_PathPrefixStripMiddleware(_ActorCredentialsMiddleware(_app)), lifespan='auto')