This is indeed a section of the Via Alpina. I have hiked some sections as dayhikes, and often thought about working in a 2-3 night trek on part of it, but haven’t managed to fit it into our Switzerland trips. We do have friends who hiked the section from Mürren to Oeschinensee with their 3-year-old in a backpack, with one overnight at Blumlisalphütte, and absolutely loved it.
My thoughts: Meiringen to Grindelwald is well worth doing, but not between Grindelwald and Lauterbrunnen and up to Mürren. That part is so well covered by multiple lifts, and even a train, that it is not fun to hike, in my view.
I have long wanted to stay at Chalet-Hotel Schwartzalp, a mountain inn which lies between between Rosenlaui and Grosse Scheidegg.
https://www.schwarzwaldalp.ch/hotel
Consider starting in Meiringen and taking your time to savor the first part, especially the Rosenlaui area, and ending the day at Schwartzwaldalp. See the trails on this area map:
https://www.schwarzwaldalp.ch/_files/ugd/6d89a4_ee3002bd783c473f9519490219e90778.pdf
The next day, continue on over Grosse Scheidegg and down to Grindelwald or, if you dislike downhill hiking as much as I do, hike from Grosse Scheidegg to the top of the First Gondola, and enjoy the views on the ride down. Walk through Grindelwald and take the train over to Lauterbrunnen and cablecar up to Mürren to spend the night. Or if you want to stay closer to the actual route and don’t mind spending some money on lifts, ride the gondola from Grindelwald Grund up to Männlichen and take the cablecar down to Wengen from there. From Wengen you take the train down to Lauterbrunnen where you take the cablecar up to Mürren. This would take you over the same terrain, with the same views, as the hike.
Or, if you feel that is taking too much of a short cut, walk or take the Postbus from Lauterbrunnen to Stechelberg,, and hike the trails into the upper valley toward Oberhornsee. There are several mountain inns along the way in this roadless valley, including Hotel Obersteinberg:
https://grindelwald.swiss/en/summer/eat-and-stay/mountain-hotels-huts/mountain-hotel-obersteinberg/
Overnight here and the following day take the trail up to Gimmelwald and then Mürren. (This is another one I have wanted to do).
Either way, from Mürren you continue toward the Sefinenfurka and Griesalp per your original plan.
