refactor(backend): phase 3 - add integration tests for config sync (partial)

Add integration test suite with isolated test environment:
- New test file: tests/import_export_sync.rs (149 lines, 3 test cases)
  * sync_claude_provider_writes_live_settings: validates SSOT sync to live settings
  * create_backup_skips_missing_file: edge case handling for missing config
  * create_backup_generates_snapshot_file: verifies backup snapshot creation
- Test infrastructure:
  * OnceLock-based isolated HOME directory in temp folder
  * Mutex guard to ensure sequential test execution (avoid file system race)
  * Automatic cleanup between test runs

Export core APIs for testing (lib.rs):
- AppType, MultiAppConfig, Provider (data structures)
- get_claude_settings_path, read_json_file (config utilities)
- create_backup, sync_current_providers_to_live (sync operations)
- update_settings, AppSettings (settings management)

Adjust visibility:
- import_export::sync_current_providers_to_live: fn -> pub fn

Update documentation:
- Mark Phase 3 as in-progress (🚧) in BACKEND_REFACTOR_PLAN.md
- Document current test coverage scope and pending scenarios

Test results: 7/7 passed (4 unit + 3 integration)
Build time: 0.16s

Next steps:
- Add Codex sync tests (auth.json + config.toml atomic writes)
- Add MCP sync integration tests
- Add end-to-end provider switching tests

src-tauri/src/lib.rs

@@ -15,6 +15,12 @@ mod speedtest;
 mod store;
 mod usage_script;
 
+pub use app_config::{AppType, MultiAppConfig};
+pub use config::{get_claude_settings_path, read_json_file};
+pub use import_export::{create_backup, sync_current_providers_to_live};
+pub use provider::Provider;
+pub use settings::{update_settings, AppSettings};
+
 use store::AppState;
 use tauri::{
     menu::{CheckMenuItem, Menu, MenuBuilder, MenuItem},