I suggest you consider the 4-day Select Swiss Pass for $337. It fully covers your travel from Chamonix to Zermatt, Zermatt to Lauterbrunnen, Lauterbrunnen to Luzern, and the Rigi round trip. Also the Schilthorn if you go on the day you arrive in or depart from Lauterbrunnen. You do not need to cover the ride from Luzern to Zurich airport with a pass because it only costs 30 CHF.
It is worth takin a full day for the Rigi excursion, as this is a case where the journey is the destination. You travel by boat from Luzern to Vitznau, then by cogwheel train to the summit at Rigi Kulm, down as far as Rigi Kaltbad, then ride the cable car steeply down to the lake at Weggis, and take a boat back to Luzern. It takes a minimum of 5 hours but it is well worth spending more time to walk, have a picnic, etc. The cost if paying full fare is 117 CHF.
https://www.rigi.ch/en/Experience-Enjoy/Summer/Summer-experiences/Classic-Rigi-Round-Trip
A Eurail pass will cover 50% of the boat and the cogwheel train, but they do not mention the cable car down from Rigi Kaltbad. A Swiss Select Pass will cover 100 % of the cost.
http://ricksteves.mytraintravel.com/pass/eurail-select-pass-2-countries-8039
Then on the day of travel between Lauterbrunnen and Luzern, you can use your travel benefit for other adventures: the Schilthorn, if you did not go the day you arrived in Lauterbrunnen, or the Ballenburg museum if interested in that, the transport museum in Luzern, an excursion on Brienzersee to Grandhotel Giessbach to see the waterfall, or other possibilities.
https://www.giessbach.ch/en/giessbach-falls-nature-park.html