In June, family of 4 traveling from Paris to Interlaken (5 nights in Lauterbrunnen) then on to Salzburg (5 nights), Rothenburg (2 nights), Fussen (1 night), Munich (5 nights). Bullet train from Paris to Switzerland? Then Eurail Pass?
The "bullet" train is from Paris to Basel, then Swiss train to Lauterbrunnen. A Eurail Pass does not work very well in Lauterbrunnen, as from Interlaken to Lauterbrunnen you just get a 25% discount, you still have to buy tickets. All your other locations are in Bavaria (even Salzburg main station) and you can travel with low cost Bayern tickets.
What are the ages of the family members? Children get reduced rates or even free. Anyway, I don't think a Eurail Pass is the way to go.
"Kids" are 18 and 20. I'll look up the Bayern pass.
Check this well regarded site for all things train:
https://www.seat61.com
Some of your trips might be cheaper if bought ahead, so check soon.
Since you will be in Lauterbrunnen for 5 days, you will want to explore the surrounding mountains by train and lift. I recommend you each buy Half Fare Cards for Switzerland at 120 CHF each, plus a 6-day Berner Oberland Pass, which is 220 CHF per person.
The covers you from Bern to Lauterbrunnen inbound, and Lauterbrunnen to Luzern on the way out (my recommendation for a scenic train ride). The Half Fare card saves 50% on tickets from Basel to Bern inbound, and Luzern to the Swiss border on the way to Salzburg.
https://www.berneseoberlandpass.ch/prices-tickets/
For daily tickets in Bavaria on regional trains, cost would be 54 EUR per day for 4 adults.
https://www.bahn.com/en/offers/regional/regional-day-ticket-for-bavaria?dbkanal_007=teaserSliderBigImage_3-1_link_laenderticketbayern