Seeing the "Unburger Patch 0.3.2" text on the main menu is a great sign. It means the mod is technically active and loading. The issue with the missing Japanese prompt and the voice options likely stems from how the game handles save data. You most likely already had a save file (or at least a settings file) on your Switch for this game before installing the mod. The game doesn't ask for the language/region if it detects that a save or configuration already exists. Also, the game saves your language settings in a separate file upon the very first startup. If that file exists, it skips the prompt you were looking for.
Here is what to do next. First, try a quick check. I'm not 100% sure what happens if you just continue, it might actually be working fine (but I doubt it will). Try starting a new game and pay attention to the very first cutscene. Look for the re-translated dialogue (specifically, right at the start, Rex should mention terms like "Eden" and "God") and listen to see if the voices are the original Japanese tracks. If both of those are present, you don't need to do anything else. You are good to go.
If that check fails (and mostly likely it will, because I'm using Japanese language slot for this translation), you need to force the game to forget its previous settings. You have two options. If you use homebrew save managers like JKSV or Checkpoint and don't want to delete your entire history: navigate to the game's save folder, find and delete only the file named "bfoption.sav", then launch the game. The language selection prompt should appear. Alternatively, if you don't mind losing progress, go to the Switch System Settings -> Data Management -> Delete Save Data, delete the save data for this game, and launch it again. You should then see the Japanese prompt.