Prove cookie injection lands, rather than assuming it

setCookie is fire-and-forget, so the import's count was what WebKit was
handed, not what it kept. A probe from inside the page reads back the
other end: pairing against Arc gave `names=tz,cids,frontend_lang` for
aputure.odoo.com, and `tz` exists only in Arc's store — so the import
demonstrably landed.

The sites still ask for sign-in. That is the far end refusing the
session, not a broken import, and the two are now distinguishable
instead of being guessed at.

src/api.ts

@@ -46,3 +46,5 @@ export const pairBrowser = (browser: string) =>
 export const testNotification = (appId: string) =>
   invoke<void>("test_notification", { appId });
 export const notificationStatus = () => invoke<string>("notification_status");
+export const probeCookies = (appId: string) => invoke<void>("probe_cookies", { appId });
+export const cookieProbe = () => invoke<string>("cookie_probe");

src/components/Settings.tsx

@@ -46,6 +46,7 @@ export default function Settings({
   const [browser, setBrowser] = useState("");
   const [pairing, setPairing] = useState(false);
   const [notifyStatus, setNotifyStatus] = useState<string | null>(null);
+  const [cookieStatus, setCookieStatus] = useState<string | null>(null);
   const [paired, setPaired] = useState<PairResult | null>(null);
 
   const groups = [...config.groups].sort((a, b) => a.order - b.order);
@@ -263,11 +264,29 @@ export default function Settings({
                       ))}
                     </select>
                   </label>
+                  <button
+                    onClick={async () => {
+                      if (!activeId) return;
+                      setCookieStatus("checking…");
+                      await api.probeCookies(activeId);
+                      await new Promise((r) => setTimeout(r, 400));
+                      setCookieStatus(await api.cookieProbe());
+                    }}
+                    disabled={!activeId}
+                    title="What the current app can actually see"
+                    className={BTN}
+                  >
+                    Check cookies
+                  </button>
                   <button onClick={pair} disabled={pairing || !browser} className={BTN_PRIMARY}>
                     {pairing ? <Spinner /> : config.settings.lastPairedAt ? "Pair again" : "Pair"}
                   </button>
                 </div>
 
+                {cookieStatus && (
+                  <p className={`${HELP} font-mono`}>{cookieStatus}</p>
+                )}
+
                 {config.settings.lastPairedAt && !paired && (
                   <p className={HELP}>Last paired {config.settings.lastPairedAt}.</p>
                 )}