fix(mcp-demux): handle Streamable-HTTP handshake correctly

LiteLLM opens a fresh MCP session per tools/call: initialize →
notifications/initialized → tools/call. Demux forwarded everything
to the WS unchanged, so:

- notifications/initialized (no id) hung forwardMcpCall waiting on
  a response id, and the 200 JSON-RPC body confused the SDK reader
  into cancelling the subsequent tools/call ("duplicate response
  suppressed").
- initialize added an unnecessary WS round-trip coupled to
  extension availability.

Answer initialize locally, return 202 empty for notifications/*,
set explicit Content-Type + Mcp-Session-Id on the handshake.

SB-48

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

src/routes/mcpDemux.ts

@@ -25,11 +25,39 @@ export async function mcpDemuxRoutes(app: FastifyInstance) {
 
       const { customer_id } = req.params;
 
+      const body = req.body as {
+        method: string;
+        params?: unknown;
+        id?: string | number;
+      };
+
+      // MCP Streamable-HTTP: notifications must return 202 empty, never forward.
+      // notifications have no id; forwardMcpCall would hang waiting for a response.
+      if (body.method?.startsWith("notifications/")) {
+        return reply.code(202).send();
+      }
+
+      // Answer initialize locally — avoids a WS round-trip per LiteLLM tool call
+      // and makes the handshake robust to transient extension disconnects.
+      if (body.method === "initialize") {
+        reply.header("Mcp-Session-Id", randomUUID());
+        reply.header("Content-Type", "application/json");
+        return reply.send({
+          jsonrpc: "2.0",
+          id: body.id ?? null,
+          result: {
+            protocolVersion: "2024-11-05",
+            capabilities: { tools: {} },
+            serverInfo: { name: "sitebridge-demux", version: "1.0.0" },
+          },
+        });
+      }
+
       if (!isCustomerOnline(customer_id)) {
         // Return a JSON-RPC error in MCP format
         return reply.code(200).send({
           jsonrpc: "2.0",
-          id: (req.body as { id?: unknown })?.id ?? null,
+          id: body.id ?? null,
           error: {
             code: -32001,
             message: "tool_unavailable: customer not online",
@@ -37,12 +65,6 @@ export async function mcpDemuxRoutes(app: FastifyInstance) {
         });
       }
 
-      const body = req.body as {
-        method: string;
-        params?: unknown;
-        id?: string | number;
-      };
-
       const callId = body.id ?? randomUUID();
 
       try {
@@ -52,6 +74,7 @@ export async function mcpDemuxRoutes(app: FastifyInstance) {
           config.mcpBridgeTimeoutMs
         );
 
+        reply.header("Content-Type", "application/json");
         return reply.send({
           jsonrpc: "2.0",
           id: callId,