I think you are going to need apartments. Teens 16 and up are generally counted as adults in most Swiss hotels, so you family could be counted as 3 or 4 adults, plus a child or two. Sometimes the “child” age even ends at 12 years. Hotel rooms sleeping 5 people are very scarce, unless they are pricy suites with 2 rooms. I did see one affordable option with a quintuple room offered on booking.com near Luzern, but it is well outside of town.
https://bellevue-luzern.ch/en/
But you can use booking.com to find large rooms, suites and apartments in each of your locations. Once identified, you can go to each individual website to book directly. Some of them will not have their own website, in which case you just use booking.com for the reservation. Their system is easy to use and communication with the host is quick and easy, in our experience..
Another option is hostel-type inns that have multi-bed rooms with bunks—-you could book a 6-bed dorm room as a private room, but that would be cramped and not particularly comfortable.