fix(copilot): handle deprecated function role and move integration test

- Add 'function' to role check in determine_x_initiator() so legacy
  function-call format is treated as agent continuation (no premium charge)
- Move test_github_copilot_billing.py from tests/llm_responses_api_testing/
  to tests/integration/github_copilot/ (real API calls belong in integration,
  not alongside mock-only tests)
- Add unit test for legacy function role

litellm/llms/github_copilot/common_utils.py

@@ -188,8 +188,8 @@ def determine_x_initiator(messages_or_input: Union[list, str]) -> str:
             # No role = agent-initiated (Responses API role-less item types)
             if role is None:
                 return "agent"
-            # assistant or tool role = agent continuation
-            if role in ("assistant", "tool"):
+            # assistant, tool, or legacy function role = agent continuation
+            if role in ("assistant", "tool", "function"):
                 return "agent"
 
     return "user"

tests/test_litellm/llms/github_copilot/test_conversation_id.py

@@ -129,5 +129,13 @@ class TestDetermineXInitiator:
     def test_string_input_returns_user(self):
         assert determine_x_initiator("What is 2+2?") == "user"
 
+    def test_legacy_function_role_returns_agent(self):
+        """Deprecated role:function must be treated as agent continuation."""
+        messages = [
+            {"role": "user", "content": "Call a function"},
+            {"role": "function", "content": "result", "name": "my_fn"},
+        ]
+        assert determine_x_initiator(messages) == "agent"
+
     def test_empty_list_returns_user(self):
         assert determine_x_initiator([]) == "user"