I realize I am posting rather close to your departure. We visited Murren two years ago and last year. There is a childrens play area at the top of the funicular in Allmendhubel. The park at Allmendhubel was finished when we were there in 2015. My 9 year old and 10 month old absolutely loved it! After an hour and half we had to drag them away kicking and screaming! Take the Funicular up and walk down the Children's Adventure trail. The Children's trail is well marked and safe to walk for small unsure footed kids. You will have to open and close gates as you enter and leave pastures. It turned cold and started to rain so be sure to carry rain gear, in our case ponchos. We saw two huge horses that were extremely friendly and goats, cows and a red fox. My 9 year old girl said it was the best day of our trip that included Milan, Rome and Venice. There are signs along the Children's Adventure trail. There is a indoor pool in Murren near the funicular. We actually stayed at Oli and Maria's in Gimmelwald. When looking at trail maps with walking times we found it best to double the travel times. We were stopping and admiring the views. The ones who set the walking times were not.
https://www.tripadvisor.com/ShowUserReviews-g188084-d6895906-r288264669-Allmendhubel_Bahn-Murren_Jungfrau_Region_Bernese_Oberland_Canton_of_Bern.html
Here is a map: http://www.momstotszurich.com/2015/04/murren-childrens-adventure-trail.html
Here is a map of all the trails in the area. http://www.jungfrau.ch/fileadmin/images/wanderwege/Jungfrau-Region_Wanderkarte.pdf
Stop in Murren and grab sandwiches at the coop or sandwich shop and plenty of water.
If going to the Shitlebahnhorn catch the earliest cable car possible be because clouds come in. We had only 30 minutes at the top when the clouds arrived. The breakfast at the top was super expensive but you are paying for the view. The clouds came in just after we sat down to eat.
The following year we stayed across the valley in Wengen, larger town and less touristy. I highly recommend the trail
33. Panorama Trail: Männlichen – Kleine Scheidegg
http://www.jungfrau.ch/en/tourism/destinations/kleine-scheidegg/experiences/
You are walking along the side of a mountain with a drop off on one side. The trail is wide enough to feel safe when walking. I saw several people with strollers make the trip but there are a few rocky sections where that would have been rough going. The views from Mannlichen was amazing with the Eiger, Mönch and Jungfrau.
Wengen has a putt putt course and some small parks, play areas and an outdoor swimming pool. From Murren, you take the train to Winteregg and then down to Lauterbrunn then back up to Wengen, the about a half mile walk to the Gondola up to Mannlichen for the start of the walk. If you are tired you can take the train from Keinne Scheidegg back to Lauterbrunn or you can walk down to Wengen and catch the train there. The cable car and train are expensive but worth it.
If I had to choose between Shchitlhorn or Mannlichen-Kleine Scheidegg, I would choose Mannlichen-Kleinne Scheidegg.
Last year we went from Lauterbrunn to Fussen by train. Pick up a sandwich and drinks before going to Neuschwanstein, according to others the restaurants at the bottom are very expensive and not very good.
Have fun.
Wade