"Would you recommend having a rental car to access adventures and fun things within 1-1.5 hours (probably our max travel time one way) or will the train system be adequate?"
You've got it backwards! Muerren is car-free, as are many of the excursions you will be taking, so a car would spend a lot of time parked in the valley below (at Swiss prices). The various trains, lifts, boats, and buses will not only be "adequate," but for many places are the only way to get there.
I agree with looking at the Berner Oberland Pass. If you're not spending time in other regions of Switzerland, and aren't going to museums, you won't get much use of the greater coverage of the Swiss Travel Pass.
Do go to the Ballenberg Open Air Museum. Make sure everyone knows that in each house, you can go into any area not specifically barred. I saw a lot of people stay on the ground floors, not realizing they could climb the stairs, go into the other rooms, etc. This museum is included in the Swiss Travel Pass; I believe there's a discount with the Berner Oberland Pass, and with either one transport from Muerren to there and back should be fully included.
I also loved the Allmendhubel - the "playground area" above Muerren that was reference above. The views mean it's not just for children!
Check out the Talmuseum, on the outskirts of Lauterbrunnen. It's about traditional lifestyles in the region. I only went because it was raining, but it's worthwhile even in good weather.
Do be sure to get Rick's Switzerland book. His chapter on this area is one of his very best.
Understand that while Swiss restaurants are very expensive (particularly if you're feeding five), Swiss supermarkets are not. There are small supermarkets in Muerren and Lauterbrunnen, larger ones in Wengen and Grindelwald (over in the next valley) and giant ones in Interlaken. These have all kinds of prepared food items, so you needen't "cook" unless you want to. Plus, during the day, you don't want to be in a restaurant - you want to be out on trails! Carrying supermarket food and drink for lunch and snacks is the way to go when outdoors.
The stores in Interlaken (Migros and Coop) are a bit like Target in the US, and are great places to buy things at non-shocking prices. When my day bag tore, I got a replacement backpack at Migros for 20 CHF.
I'm very jealous for you and your kids - you'll all have a great time!