fix(proxy): allow trailing slash and sub-paths on MCP inference routes

`/mcp/{subpath}` compiles to `^/mcp/[^/]+$`, so `/mcp/<server>/` and
`/mcp/<server>/mcp` are not recognised as LLM API routes and fall through
to the admin-only branch, producing a misleading key/user/team error.

Use the `:path` placeholder already used by `mcp_management_routes`.

Fixes #36531

litellm/proxy/_types.py

@@ -472,7 +472,7 @@ class LiteLLMRoutes(enum.Enum):
     mcp_inference_routes = [
         "/mcp",
         "/mcp/",
-        "/mcp/{subpath}",
+        "/mcp/{subpath:path}",
         "/mcp/tools",
         "/mcp/tools/list",
         "/mcp/tools/call",

tests/test_litellm/proxy/auth/test_route_checks.py

@@ -462,6 +462,8 @@ def test_virtual_key_llm_api_routes_rejects_mcp_multi_segment_admin_subpaths(
         ("/mcp", "POST"),
         ("/mcp/", "POST"),
         ("/mcp/my-server", "POST"),  # matches the /mcp/{subpath} pattern
+        ("/mcp/my-server/", "POST"),  # same server, trailing slash
+        ("/mcp/my-server/mcp", "POST"),  # streamable-http transport sub-path
         ("/mcp/tools", "GET"),
         ("/mcp/tools/list", "POST"),
         ("/mcp/tools/call", "POST"),