Bettmeralp is nice and has amazing mountain carts with no lines. We road them over and over again without another person there. There is also nice hiking along (or ON) the Aletsch Glacier. Your family might enjoy that unique experience. Again I want to clarify that you’re doing later in June as some places really won’t fully open up for the season until mid-late June.
Personally with everything you’re trying to accomplish, I’d just give Luzern 1 night and head into the mountains from there. Old Town can be seen in a couple hours and is a good way to keep yourself up after a long flight so that you don’t go to bed too early from jet lag. Land, head to Luzern, drop off your luggage, walk around for a couple hours, have dinner, head to bed and then go to the mountains the next morning. You’ll likely wake up a bit later than normal from the jet lag, which works well to make it a travel day into the mountains. With only 13 nights and this much mountain territory to cover, you need the extra buffer days for weather in the mountains. It’s not always perfect and every extra day you get ups your chances of enjoying the best things there are to offer there.
There are many family friendly huts in the Berner Oberland, but if you want to do another one from your list then you should add on another night to that area. In regards to your list - Glecksteinhutte is a 3000 ft ascent, Baregg is one of my least favorite hikes I’ve done in the area and Rugghubelhutte is in a completely different area, so the logistics seem complicated to add to your list. Lobhornhutte was the other one I was going to suggest for you. Or just pick Bettmeralp for a non touristy option. Saas Fee, Bettmeralp & Mürren are all car free, so in a way you get the mountain hut experience without the requirement to hike there with your stuff and you can do great day hikes to these huts, but then sleep somewhere with a hot shower.
Going down to Stechelberg to get to Obersteinberg isn’t really going to help you that much, as you still have to get down from Rotstockhutte to Mürren to take the cable car and then you have to gain 2800 ft from Stechelberg. Going directly from Rotstockhutte to Obersteinberg takes you a different way altogether gaining 2400 ft instead and cutting off a couple miles of hiking. You have to go an extra 1500 ft of downhill but it is shorter and less gain. Either way, once you’ve gotten to Tschingelhorn you’ve pretty much done all the hard part, so it’s not really a mid way break. I will also note that my husband & I are extremely active hikers in the area all summer, every summer, and Obersteinberg is not a very “fun” hike in our opinion. I describe it as a stair climber in the woods for a good portion of it. The reward is great, but to get there isn’t exactly my favorite :) If a 6 mile hike (2400 ft gain/3300 ft loss) feels like too much to you, then Trachsellauenen is 5.4 miles 3000 ft of loss but only 350 ft of gain. There isn’t really a way to make a hut to hut less than that in that area, so your other option is to do Rotstockhutte > stay in Mürren > on your way out, store your luggage in Lauterbrunnen > bus + cable car up to Sulwald > 1400 ft gain hike to Lobhornhutte. In that instance you’d want to stay somewhere with a hot shower before flying home. You could stay up at Arva on Oeschinensee for a couple nights. It has the feeling of a mountain hut as you take a short walk to get there, but it’s directly on the lake with hot showers, high quality food and the benefit of having a really popular and busy lake to yourself when the cable car isn’t running. We stayed up there for a week a few summers ago and I think your kids would love the ability to go swimming right outside your place, take the row boats out, hike up to the viewpoints, etc. It’s like a boogie mountain hut :)
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