We are traveling also from Strasbourg to Mürren. For better or worst this is what we decided to do (but please point out the errors of my ways, if there is a better way):
We already bought Strasbourg to Basel.
Because we already have been to the Bernese Oberland area, we are not looking to go up the Jungfrau top of Europe. 60 days in advance, we would like to purchase the super saver tickets (only good for that specific train):
https://www.sbb.ch/en/travelcards-and-tickets/tickets-for-switzerland/supersaver-tickets.html
Basel to Mürren is 52.80 CHF (26.40 with the half fare card).
Children would be free with the half fare card but if we decide not to buy it, I can buy for my 6 years old a children's co-travel card for 30 CHF, that would give her free access to all transport as long as she is with a paying adult.
A previous poster mentioned if we were to spend at least 240 CHF on transport, then it would be worth it to get the half fare travel card. In the past I got it because I did not want to deal with the headache of adding everything and we took some of the most expensive funiculars.
But I think this year we will go without. Round trip Basel to Mürren with the children's co-travel card if we can get the supersaver is 135.60 CHF.