Good morning! I had recently contacted the local bike shop w/ a similar question and below was there helpful response:
There is a few mountainbiking trails in the area, but not too many. The area is quite steep, so you should have been riding mountainbikes before.
The easier one, and only one that is paved is on the valley floor from Lauterbrunnen to Stechelberg. You can also ride from Grütschalp to Mürren via Winteregg. The ride up to Allmendhubel is possible but very very steep and you are not allowed to take the bikes on the Allmendhubelbahn. The cable cars up till Mürren will let you take your bike with, for an additional charge.
You can ride from Mürren down to Gimmelwald, the path is paved and it takes about 10 minutes.
The other option, people normally do, is to ride from Mürren down to Lauterbrunnen via Winteregg. Just be aware that it’s a loose gravel road which is pretty steep downhill. I would only recommend you going, if you are an experienced mountainbiker, because it is certainly not easy at all and constantly downhill, no flat areas.
In case you have seen the Rick Steve video/article, it is very misleading and not correct, lots of people getting trapped with it.
The other option would be to rent bikes in Lauterbrunnen, there is a big bike shop there, and you can ride through the valley on the valley floor, which is flat, but very scenic and beautiful, especially in spring with all the waterfalls.
Prices are the same, full day rental for a bike including a helmet is 40,- half a day would be 30,-
Also if you rent down there you don’t need to pay for a cable car ticket for your bike, to bring it back up.