Ok, sounds like you can handle the trains then! It will be a fun and memorable adventure and the good thing is that if you miss a train connection for some reason, you’ll catch the next one. No big deal :) We spend our summers living in Murren and my young nephews came to visit us last summer for 2 weeks (ages 3 & 1) and we had the best time exploring the Berner Oberland with them. There are fun open air historic train rides to do that kids love (Schynigge Platte, Brienzer Rothorn) and so many family activities that you could spend your whole time there and not run out of activities to do.
For September I’d definitely prioritize getting lodging ASAP. Especially for popular places like Murren. As I said, we live there for the summer and have family coming to visit us this September so I’ve already seen that stuff is getting pretty booked up.
For lodging in Murren look at e-Domizil, Airbnb, VRBO & https://chalet.myswitzerland.com/holiday-rentals/ or if you have a decent budget I know of a larger penthouse apartments that would be great for a family and is right next door to the soccer field, pool and Allmendhubel funicular up to the playground.
I can help you more with specifics/logistics of Murren if you decide on that specifically. There is also some train passes for the area that would be of help to you depending on how long you stay.
Other places to look into as an additional base - Lake Luzern (Luzern or we like the quieter/prettier side over by Fluelen, etc), Lake Geneva (we’ve stayed in a small family friendly village called St.Prex where we had a waterfront house next to a little beach where kids can swim and was a 5 minute walk to the train station to visit places like Montreux, etc) — other car free villages to look into - Bettmeralp, Zermatt….
I think you’d really love doing the car free villages with kids. It’s a unique experience and makes for an overall relaxed experience with kids. With the train system you could still do longer day trips to some of the other places of interest, but I think staying in car free villages as your “home” is perfect for a young family and will be a really special experience for you.