Swiss Pass use from Lucerne to Milan

No, the easiest way is to stop at the ticket office, show you STP and buy a ticket to Milan. They will only charge you for the Italian portion of the trip. There is 1 direct train per day at 12:18, the rest require a connection at Arth-Goldau.

Or you could opt to take the Gothard Panoramic Express that starts with a lake boat to Fluelen, then transfer to a train over the old route that uses a shorter tunnel than the 25 mile long new Gothard Base Tunnel. That train will drop you in Lugano, where you get another train to Milan. I think this train has a reservation fee that has to be paid and only the 1st class section has panorama cars. There are other trains on this route from Luzern with more train changes, no panorama cars, and no reservation fees.

No special status. A Swiss Pass will cover you up to the last Swiss station (which, on this route is Chiasso, not Lugano). You then need to buy an Italian ticket (on the Trenitalia website: https://www.trenitalia.com/tcom-en ) from Chiasso to Milano.

The official Swiss Pass map shows what is covered (map: https://www.sbb.ch/content/dam/infrastruktur/trafimage/karten/karte-sts-pass-gueltigkeit.pdf ). You will see Chiasso as the last red station bottom, right of centre.

First you have to decide which trains to get. Look up train times on the SBB website: https://www.sbb.ch/en You need to do this on the SBB website, as Trenitalia does not not show Swiss trains.
Once you have decided which train to take, click on the selected option to see the individual trains, and click on '+' to show all the stops. Look what time this train passes through Chiasso.

Now, on the Trenitalia website, buy a ticket on this train (same date, same time, same train ID) from Chiasso to Milano. This will be a "print-at-home" ticket.

You don't have to get off the train at the border. The train crew changes at Chiasso. Just put your Swiss Pass in your bag (souvenir), and get out your Italian ticket to show the Italian ticket inspector when (s)he comes around.

Thank you to both Sam and Chris who responded to my query. Each of your responses was helpful to my decision-making.