Finally, based on Gen solution, I did this:

1 - Open the System Settings and enter an unwanted preference (for example, Mouse2 if you want Mouse3 in the end.

2 - Open the ~/Library/Containers/com.apple.Desktop-Settings.extension/Data/Library/Preferences/com.apple.symbolichotkeys.plist file with a pList editor, and note the keys and their values (or take a screenshot).

3 - Return to the System Settings and enter the correct preference (Mouse3 in my case).

4 - Open again ~/Library/Containers/com.apple.Desktop-Settings.extension/Data/Library/Preferences/com.apple.symbolichotkeys.plist file with a pList editor, and paste the modified keys (comparing them with the previous ones) into the ~/Library/Preferences/com.apple.symbolichotkeys.plist file.

5 - Save this file.

Since the ~/Library/Containers/com.apple.Desktop-Settings.extension/Data/Library/Preferences/com.apple.symbolichotkeys.plist file does not seem to be saved between each session, the idea is to locate the affected keys and paste them into the ~/Library/Preferences/com.apple.symbolichotkeys.plist file, which seems to be correctly saved.