If the landlord does not want you to modify the existing structure then your options are limited. The simplest would be to get by with the curtains.
If you were really handy it may be possible to build an alternate door frame that gently slides into the existing frame and then is just wedged into place using some thin taper wedges. The new door would attach to this slip in frame.
To make this work would likely require a threshold piece all along the bottom that would sit on top of the existing floor covering.
This could end up pretty ugly or looking rather nice based upon the craftsmanship applied. Two potential issues would be the reduced height in the opening and the threshold piece could be a potential trip hazard.