Can you fish in the beautiful Weisse Lütschine?

Yes, but with a permit. There are lovely lake or brown trout there. Only from Mid-March to end September. You must be able to agree to fishing by the methods stipulated by the Bern Fisheries department and it is quite costly.
 
New regulations from 1 January 2020 say that a Bern fishing license can in principle be obtained by anyone - with or without a residence in the canton.  Although you can buy online, it will be posted otherwise you can buy from Tourism office in Lauterbrunnen.
 
The fee system is a bit complex, but basically a weekly permit is around 100CHF. It is personal and non-transferable and you need to report back the catch statistics. Each person who is fishing needs a license even if they share the rod except for kids under 9 if their accompanying adult has a permit.
 
You can find information here: www.vol.be.ch/vol/de/index/natur/fischerei/angelfischerei/patente/preise.html

​While it could all sound a bit hard, the good news is fishing without licence is allowed in Lake Brienz and Lake Thun.
 
Of course there are some rules to observe such as fishing from the shore with a single rod and one single barbless hook is allowed. There are limits to numbers and sizes of course.