You might check with local charities concerning where they hold annual stair climb events, and how they obtained permission.
These events commonly are run up the stairwells of relatively tall building. In Seattle, for instance, one such ran up the Columbia Center fire escape stairwells (now under a different name that I've forgotten), seventy-six floors or ...
Have you checked out nearby public buildings? You're looking for stairs that are accessible (i.e. not the fire escape), but not crowded (not the showy staircase in the lobby).
Possible candidates: the library, multistory department stores and malls, hospitals.
The solution depends on the design of the device. If the cover is part of the electric circuit, it gets more complicated.
To find out if this is the case: does the device work when you insert the battery and hold it in place with your finger? Then the cover isn't part of the electric circuit, and this should work:
Find some non-corrugated cardboard.