A "Cornhusker" ? I have relatives near Omaha. Hope to see the eclipse with them later this month.
Welcome to My Swiss Alps.
<<"From Bern we will visit, Lucerne, Murren, Montreux, and then back to Geneva, staying roughly 2 days in each place. ">>
On the intercity routes for the trip you describe, there will be no private lines of the type that require an extra charge if you have Swiss Travel Pass, which is the kind that covers almost all costs.
If you make mountain excursions, such as to the Rocher de Naye in Montreux, you may find a surcharge. If you go beyond Mürren, Wengen, or Grindelwald, you will find surcharges. For example, to the Jungfrau - that journey is covered as far as Wengen , since it is an "intercity" journey, even though the same train continues on up to Kleine Scheidegg, because it then become an excursion train after it leaves Wengen. Often, such trains are not covered at all by Eurail passes.
For more details about travel passes, I defer to the experts on that subject- the moderators of the Forum, Arno, Annika, Lucas.
Have you discovered the timetable?