Plugin sync can now run in the background during startup and idle time, under a time budget

What

• A new idle-time plugin refresh path races the plugin sync against the turn's remaining time budget; if the sync runs long, it still finishes afterward rather than being abandoned. This is controlled by the tengu_ccr_plugin_refresh_in_idle_enabled setting.
• Startup can now kick off plugin sync in parallel with the message loop starting, tracked against a remainingSyncBudgetMs derived from the CLAUDE_CODE_SYNC_PLUGIN_INSTALL_TIMEOUT_MS setting, and flags when the sync outlives that budget.
• The surrounding loop now also records timing telemetry (first_message_handle_ms, first_message_handle_cpu_ms, first_message_handle_loop_lag_ms) to help track how these overlapping operations affect responsiveness.

Why

By letting plugin and MCP syncing happen alongside startup or idle time instead of blocking it, Claude Code can respond faster while still completing the sync in the background, with a budget so it doesn't take over indefinitely.