Can you fish in the beautiful Weisse Luechine?

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 100 CHF. 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. There you can find your own nook around the lake, spread a blanket on the grass and enjoy a picnic while you wait for your catch.
 
Of course there are some rules to observe such as fishing from the shore with a single rod and only one single barb- less hook is allowed.  Naturally limits to numbers and sizes apply, so best to investigate the above website further.