Show Offline when the server is unreachable, and document the pilot changes

The terminal's badge only looked at navigator.onLine, so it said Online
while the server was down. It now also reflects whether the last sync or
health check reached the server. The cart badge no longer covers the
product name.

docs/PILOT_READINESS.md records what changed against the PRD review, the
decisions taken (offline sales may take stock negative, transfers may
not, server price wins but differences are flagged), rollout notes, and
what is still open.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

src/lib/offlineQueue.ts

@@ -217,6 +217,11 @@ async function doFlush(owner: QueueOwner): Promise<FlushResult> {
   let synced = 0;
   let rejected = 0;
   let offline = false;
+  if (pending.length === 0) {
+    // Nothing to send: still find out whether the server is reachable, so
+    // the terminal's Online/Offline badge is truthful.
+    offline = !(await fetch("/api/health", { cache: "no-store" }).then((r) => r.ok).catch(() => false));
+  }
   for (let i = 0; i < pending.length; i += MAX_SALES_PER_REQUEST) {
     const chunk = pending.slice(i, i + MAX_SALES_PER_REQUEST);
     try {