feat: migrate Claude common config snippet from localStorage to config.json

Migrate the Claude common config snippet storage from browser localStorage
to the persistent config.json file for better cross-device sync and backup support.

**Backend Changes:**
- Add `claude_common_config_snippet` field to `MultiAppConfig` struct
- Add `get_claude_common_config_snippet` and `set_claude_common_config_snippet` Tauri commands
- Include JSON validation in the setter command

**Frontend Changes:**
- Create new `lib/api/config.ts` API module
- Refactor `useCommonConfigSnippet` hook to use config.json instead of localStorage
- Add automatic one-time migration from localStorage to config.json
- Add loading state during initialization

**Benefits:**
- Cross-device synchronization via backup/restore
- More reliable persistence than browser storage
- Centralized configuration management
- Seamless migration for existing users

src-tauri/src/lib.rs

@@ -511,6 +511,8 @@ pub fn run() {
             commands::get_init_error,
             commands::get_app_config_path,
             commands::open_app_config_folder,
+            commands::get_claude_common_config_snippet,
+            commands::set_claude_common_config_snippet,
             commands::read_live_provider_settings,
             commands::get_settings,
             commands::save_settings,