Zermatt-Chamonix-Lauterbrunnen (Rent Car or Train?)

Easy answer, train. A car is a burden not a convenience where u are going.
I have traveled your areas myself by train, perfect.

And Zermatt is car free, you have to leave it in the paid lot in Taesch and take the train the rest of the way. They do have electric taxis and buses in Zermatt to get around town.
Although Lauterbrunnen has car traffic, the villages above of Muerren and Wengen, where you want to be, are car free as well. You have to take a train or cable car to get there.

A big reason to go by train is that renting a car in one country and dropping it off in another is very expensive. Even if budget is not an issue it may pop your eyes out of their sockets when you get a quote.

A bigger reason is that train travel in Europe is so easy it must be experienced. It is a joy to sit back and eat, nap, visit, watch the scenery. The driver gets short-changed IMO.

FYI, one day stays in a mountainous area can be disappointing. The Matterhorn and Mont Blanc may choose not to show themselves. I would drop Chamonix as an outlier that will take a long time to travel to and from and add that time (2 nights?) to the other two locations: 3 nights Zermatt and 4 in the Jungfrau Region.

Note it is better to state your itinerary in number of nights. Gives us a better idea of your pace.

I couldn't agree more with Laurel. Lauterbrunnen in absolutely stunning, when the visibility is good. I was there four days and to be honest, much of the time we couldn't see the mountains.

My rule of thumb for the Lauterbrunnen area is that 1 out of 3 days will be overcast, misty, or rainy.

For Zermatt, 50/50 chance.
So, plan accordingly.

Since most of your train travel is within Switzerland, consider a one-month Half-Fare Card or a Swiss Travel Pass (probably for either 4 or 8 consecutive days). Either one applies from Domodossola northward; a ticket from Milan to that point costs only about $20 in 2nd class. You'll need another, separate ticket from the Swiss border to Innsbruck.

1 night in Chamonix, 1 night in Zermatt and 2-3 in Jungfrau region sounds way too rushed for me.
Drop Chamonix and the train then works much better and your plan is far less rushed.

Listen to others on the weather/cloud/overcast issue.
Being in Zermatt for 1 day and a big cloud has settled in front of the Matterhorn kind of ruins the expensive experience.
You will be very lucky if half your days you are able to see the mountains clearly.