I downloaded all the dubbing files, but the installer returned with the error message saying "This patch doesn't support your version of the game, please download a new installer".

Is the dubbing available for the complete edition or only for the GOTY edition?
Is there anything I can do to install dubbing on my game?

If you use the GOG Galaxy => Manage installation => Configure there is an option to choose language - there you can choose polish version as an option, then in settings the dubbing can be chosen in game (although should be turned on by default when chosen during installation) - there is no need to add any files by hand as far as I am aware. That being said I did not try to install without Galaxy. You might need to change the audio language to polish in game (before starting new game/loading, in the main menu).

The polish dub was available ever since the lunch so it is definitely available for all versions. I have also tried the polish dub in the Ultimate version it works in the latest version (v4.04).