Some progress today.
The beige plastic bridge broke in half. Replaced it with a butchered old Höfner ebony bridge. That alone has probably doubled the bass's value ;-)
Messed with the wiring a bit, now I have bridge pickup and bridge+neck. Can't solo the neck pickup yet.
Shimmed the neck and could set the action to medium-low.
Here are the bridge, the Soviet bloc harness, and the truss rod access under the neck-heel plate.
From what I find, it's a Kavkaz model Bas-2, built in the Ros-Muz-Prom factory in Rostov-on-Don, and originally cost 165 Rubles, which was a lot of money (almost two average monthly salaries in the USSR). Today, btw, 165 Rubles = $3.