feat(client): show gateway peer in startup status

The relay welcome now carries the gateway peer id, but the first Windows client
startup line still only showed whether a gateway was connected. Surface the peer
id there when it is known so manual MVP testers can record it even before or
without relying on a later lifecycle catch-up line.

Keep the older yes/no text when a gateway is connected but the peer id is not
available, which preserves useful output for older welcome payloads.

Test Plan:
- cargo fmt --check
- cargo test -p lanparty-client-win gateway_status
- cargo clippy -p lanparty-client-win --all-targets -- -D warnings
- git diff --check
- git diff --cached --check

Refs: MVP manual validation

crates/lanparty-client-win/src/main.rs

@@ -193,7 +193,10 @@ async fn main() -> Result<()> {
         session.welcome().room_id(),
         session.welcome().effective_tap_mtu(),
         session.welcome().mode(),
-        yes_no(session.welcome().gateway_connected())
+        gateway_status_label(
+            session.welcome().gateway_connected(),
+            session.welcome().gateway_peer_id()
+        )
     );
     #[cfg(windows)]
     let relay_route_pin = match pin_relay_route_before_tap(session.config().relay_addr().ip()) {
@@ -755,6 +758,14 @@ const fn yes_no(value: bool) -> &'static str {
     if value { "yes" } else { "no" }
 }
 
+fn gateway_status_label(gateway_connected: bool, gateway_peer_id: Option<u32>) -> String {
+    match (gateway_connected, gateway_peer_id) {
+        (true, Some(peer_id)) => format!("yes (peer {peer_id})"),
+        (true, None) => "yes".to_owned(),
+        (false, _) => "no".to_owned(),
+    }
+}
+
 fn optional_label<T: std::fmt::Display>(value: Option<T>) -> String {
     value.map_or_else(|| "unknown".to_string(), |value| value.to_string())
 }
@@ -1243,6 +1254,13 @@ mod tests {
         );
     }
 
+    #[test]
+    fn formats_gateway_status_with_welcome_peer_id() {
+        assert_eq!(gateway_status_label(true, Some(7)), "yes (peer 7)");
+        assert_eq!(gateway_status_label(true, None), "yes");
+        assert_eq!(gateway_status_label(false, Some(7)), "no");
+    }
+
     #[test]
     fn formats_missing_client_diagnostics_as_unknown() {
         let diagnostics = ClientDiagnostics::new(