The SBB app shows I need to travel to Stechelberg (Schilthornbahn) then to Gimmelwald (Schilthornbahn) cable cars to get to Murren. Should I go on and book this trip since BLM train is supposed to be closed too?
No need to "book". You can buy a ticket wen you get there if you want. Or buy with the app.
The Lauterbrunnen - Grutschalp cablecar, and the Grutschalp - Murren railway are indeed currently closed for maintenance. This is always the case for a couple of weeks in spring. But Murren can be reached via Stechelberg and that is indeed exactly what the SBB website and app will propose.
So you can still go. Just buy a ticket from wherever your trip starts to Murren, and you are good. Note however that the areas around the cable car stations in Stechelberg and Murren are currently a bit of a building site as well.
The gondola is used everyday all day to transport people who live and work there (including cows), so it's mass transportation, not like booking an airline. I dont think you can buy a ticket ahead of time that reserves a (non-existent) seat. The change at Gimmelwald to go further up is a short walk to the next gondola. It will all be self-evident when you see it.
the short walk mentioned by stan is really short - maybe 10 feet...