fix(proxy): sync live config tokens to database before takeover

When proxy takeover is activated, tokens are replaced with placeholders
in live config files. However, the proxy reads tokens from the database,
not from live files. If the user's token only exists in the live config
(e.g., manually added), the database won't have it, causing auth failures.

Changes:
- Add sync_live_to_providers() to sync tokens from live configs to DB
- Add update_provider_settings_config() DAO method for partial updates
- Use "PROXY_MANAGED" placeholder instead of empty string to avoid
  "missing API key" warnings in Claude Code status bar
- Integrate sync step into start_with_takeover() flow before clearing tokens

The new takeover flow:
1. setup_proxy_targets()
2. backup_live_configs()
3. sync_live_to_providers() <- NEW
4. takeover_live_configs()
5. Start proxy server

src-tauri/src/database/dao/providers.rs

@@ -483,6 +483,26 @@ impl Database {
         Ok(targets)
     }
 
+    /// 更新供应商的 settings_config（仅更新配置，不改变其他字段）
+    pub fn update_provider_settings_config(
+        &self,
+        app_type: &str,
+        provider_id: &str,
+        settings_config: &serde_json::Value,
+    ) -> Result<(), AppError> {
+        let conn = lock_conn!(self.conn);
+        conn.execute(
+            "UPDATE providers SET settings_config = ?1 WHERE id = ?2 AND app_type = ?3",
+            params![
+                serde_json::to_string(settings_config).unwrap(),
+                provider_id,
+                app_type
+            ],
+        )
+        .map_err(|e| AppError::Database(e.to_string()))?;
+        Ok(())
+    }
+
     /// 添加自定义端点
     pub fn add_custom_endpoint(
         &self,