I have a different take on this from Laurel. I think a 4-day Flex will be better (but I make no judgment as to how that might compare to a Half Fare Card for the adults and free Family Card for the children).
A 4-day BO pass costs 250 CHF, but there is no Family Card included for free, so you will pay an additional 30 CHF per child for them to travel with you on the pass. Total cost 310 CHF, exactly the same as a 4-day Flex Swiss Travel Pass.
The Swiss Travel Pass fully covers the Schilthorn and gives you 50% off the cable car from Maennlichen to Kleine Scheidegg and the train from KS back down to Wengen.
The BO Pass fully covers the Maennlichen lift and train from KS to Wengen, but you only get 50% off the Schilthorn. So there are opposite. I did not check prices but I suspect the better benefit is the Schilthorn ride.
The BO pass only covers from Bern, so you will have to pay for all four people from Basel to Bern (41 CHF) and from Bern to Zurich (56 CHF each to the airport). I honestly do not know if kids pay less as we have never bought childrens’ tickets in Switzerland, we always use a pass.
Either one will only cover four days, so you will have to pay full price tickets on one “uncovered “ day. The BO pass is for consecutive days only, so your uncovered day will be either you first or last day: the long travel day from Basel to Mürren or from Mürren to Zurich Flughafen. The less expensive of these would be the trip from Basel. I need to look up the price but it will be a lot for four people, even if there is a discount for children, which I do not know about.
With the Flex Pass, you can choose to use it to cover the most expensive four days and buy tickets for the one least expensive day, which would be your Interlaken day. Mürren to Interlaken is 18,80 each way, so you need to multiply that by 4X for the adults and 4X for whatever the child price is. It will still be cheaper than Basel to Mürren for the four of you. But if I were you, I would re-think that day’s hike and choose something close to Mürren, like the walk (or funicular ride) up to Allmendhubel and then the Children’s Adventure trail or other trail down from there. I say this knowing nothing about the caves at Interlaken (never heard of them) but I am not a cave explorer.
In the end, I believe the Swiss Travel Pass 4-day Flex will cost you less.
And now I need to go look at child prices on SBB to be more helpful.