Lucerne to Zermatt

Given you are planning a lot of train travel you are probably either going to get the Swiss Travel Pass or the Half Fare Card I assume.
(I hope you didn't just buy Glacier Express tickets without researching this).

And so the train is quite probably going to be the best option for you. Renting a car just to go from A to B is not sometihng people do here, so it will come with extra costs etc...

Also with only on day in Zermatt the Matterhorn might be in the clouds and a long trip and train costs might be wasted .

You will actually get to Zermatt easier by train. 3 1/4 hours if you leave on the hour with connections is Bern and Visp. You could buy Saver day Passes today for 52 CHF per person if you do not have Half Fare Cards, which is a savings of 70 CHF per person over buying point-to-point tickets.

My husband and I did that same itinerary last year of staying in Luzern, and then taking the train to Zermatt. Zermatt is charming, and we enjoyed walking through the car-free village, as well as going up the Matterhorn by the 3 different routes, including train, gondola and cable car, for glorious views, a movie, and outdoor cafe along the mountain. Therefore, I suggest staying more than one night. Since cars are not allowed in Zermatt, you would need to drop a rental off, and then make arrangements to get to into the town..