fix: make the integration work against a Claude proxy

- Treat an empty tool argument string as no parameters instead of raising
  JSONDecodeError; the proxy sends "" for tools like the timer status
  (upstream issue #3).
- Do not replay earlier web searches in the history. The chat log keeps only
  their status, and a search without results makes the model disown the
  answers it based on it and search again (upstream issue #4).
- Look up the home location for web search with the subentry's model instead
  of gpt-4o-mini, and save the options without city/region if the lookup
  fails (upstream issue #5).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

tests/test_conversation.py

@@ -231,19 +231,6 @@ async def test_conversation_agent(
 @pytest.mark.parametrize(
     ("description", "messages"),
     [
-        (
-            "Test function call started with missing arguments",
-            (
-                *create_function_tool_call_item(
-                    id="fc_1",
-                    arguments=[],
-                    call_id="call_call_1",
-                    name="test_tool",
-                    output_index=0,
-                ),
-                *create_message_item(id="msg_A", text="Cool", output_index=1),
-            ),
-        ),
         (
             "Test invalid JSON",
             (