How to “lock” a bathroom door without drilling or spending too much money?
I'd be inclined to replace the locking mechanism with a new similar one. Direct replacement. Always change back at end of lease. Who's to know? Not expensive.
Otherwise a notice outside the door - 'Do not disturb', or get used to whistling while engaged...
Or - a length of wood that spans from the door to the opposite wall, or some article of furniture closer
BUT - bear in mind it's not too clever to have a bathroom door that cannot be opened in an emegency, from outside. That's why a lot of folk use the sort of lock with a screwdriver hole on the outside of the door. As a landlord, it's his responsibilty to provide stuff like that - not ony probably fulfilling legal obligations, but sensible ones too.