fix: set Claude sessionKey in cookie jar instead of raw header

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

src/providers/claude.py

@@ -32,10 +32,10 @@ class ClaudeProvider(BaseProvider):
                     "Run 'python -m src.main auth' to configure it."
                 ),
             )
-        # Set cookie header; never log the key value
+        # Set sessionKey in the cookie jar (not as a raw header string)
+        self._session.cookies.set("sessionKey", key, domain="claude.ai", path="/")
         self._session.headers.update(
             {
-                "Cookie": f"sessionKey={key}",
                 "Referer": "https://claude.ai/",
                 "Origin": "https://claude.ai",
             }