By using precise scientific approach and materials (i.e. a bit of foam board and a few nails) I "designed" a mechanical linkage which can move between two required positions.

Based on that I added a bunch of holes in the keyboard case and a few holes on the levers to try out different lever lengths and fixing positions to get something similar to the prototype.

Unfortunately, as I assumed but for some reason hoped it won't happen, the mechanisms on both sides need to be coupled for it to work properly.

After some more time in FreeCAD, I added the coupling shaft (i.e. M4 threaded rod) to the front side to drive two levers together. Motion from servo was transferred using two gears with same number of teeth.

This worked well from the mechanical side but unfortunately (again), to move the whole deck the servo required a bit more current than the battery could provide (the battery protection kicked in). And since it can only move 0-180° I couldn't add gears to reduce the required torque.

The (partial) solution was to replace the servo with a geared DC motor. I also added the hall sensor inside and mounted two magnets on the bottom part to detect the end positions and stop the motor. The one I have isn't perfect as the gearbox on it doesn't have the required ratio so even with additional 3D printed 1.85:1 gears (higher ratio probably wouldn't fit) it still doesn't have enough torque to start lifting the screen. But with just a bit of manual help it manages to do it. There's quite a bit of backlash and the close operation is quite fast so I might replace the motor with the slower one at some point.