Staying in Lucerne for 3 nights, would like to take the Bernina Express

No, you cannot do it in one day.

The Bernina route (it is a route with one train per hour, not just the Bernina Express) runs from Chur, over the Alps, to Tirano in Italy. It takes over 4 hours. It is best done as your way of getting from Switzerland to Italy, or vice-versa.
If you want to double back, that is 9 hours travel, to and from Chur.

The drive from Luzern to Chur is 1h40, see here: https://www.viamichelin.com/web/Routes?departure=Luzern%20(Lucerne)%2C%20Switzerland&arrival=7000%20Chur%2C%20Switzerland

The best option is to use the Bernina route as your method of crossing the Alps between Switzerland and Italy, one way.
Second option to stay somewhere along the Bernina route.
Third best option to stay in Chur.

Why are you hiring a car? Why not train from Luzern to Chur?
And, where are you returning this car? Are you aware that hiring a car in one country and returning it in a different one has a large surcharge.

Thanks so much for the suggestions.
Yes. Looks like a bad idea to travel from Lucerne to Chur just to take the Bernina Express..

So we will land in Zurich, pick up rental car ( already booked for 15 days) drive to Chur, stay for two days, take the Bernina, return, and next morning onward to Austria & Bavaria for the next 12 days. Return to Zürich via Salzburg and Lindau ( through the Alpenstrasse). Will stay in Baden for 2 nights, drive to Lucerne for a day, return to Zürich, return rental car and fly back home. Hope this is a good itinerary for a not -so -young couple, for late Summer? Focusing only on Zürich, Chur and Lucerne for a first trip to Switzerland.

Chur, stay for two days, . . . . . Will stay in Baden for 2 nights, drive to Lucerne for a day, return to Zürich,
Focusing only on Zürich, Chur and Lucerne for a first trip to Switzerland.

• Chur - OK as you want to do the Bernina Route so much. Much better to stay somewhere smaller along the route (Pontresina is a favourite, others possible). That would mean finding (paid) parking in Chur.
• "Will stay in Baden for 2 nights" - Which Baden (there is one in Switzerland, one in Germany and one in Austria)? None of which I would recommend to a visitor.
• Luzern - good choice.
• Zürich no, do not spend time there. Go direct to the airport.

Other points:
You don't say where you are coming from. If your Driving Licences are not from an EU country, each licence will need an IDP (International Driving Permit) to drive in Austria.
The car will need a Vignette (toll payment sticker) to drive on Austrian Autobahns. Buy at the border. Any car hired in Switzerland will already have a Swiss Vignette. German Autobahns have no tolls.
Always check hotels have parking. Many don't and even if they do it will usually cost.

Flying in from US, but have traveled in Europe, mostly driving through Italy, Spain, Portugal, Slovenia, Austria and Croatia in the past.

Baden in Switzerland, since its close to Zürich, and Airbnb homes here come with parking.

The reason we thought of Zürich is to visit a chocolate factory, and maybe a cheese factory as well.

Baden (Switzerland) is an industrial town. I can think of many places I would rather stay.
There are both chocolate factories and cheeseries you can visit in Switzerland, but Zürich is not where I would look for them. I suggest an internet search.

There are also cheeseries in Vorarlberg (the bit of Austria nearest Switzerland, which you will drive through on your way to Mittenwald).
Try googling: Schaukäserei Vorarlberg

Thanks so much Chris. We will do that.
Know of any Chocolate factory that we can visit en route to Mittenwald from Chur?