Hi there! My name is David. It’s a pleasure to help you today.
Regarding your question, unfortunately, in Windows 11 it is not possible to use Spotlight on one screen and set a custom background image on another, because the Spotlight setting applies to all screens.
As additional information, you can use the Feedback Hub App to send your comments or suggestions so that the feature you require, could be implemented in the future. You can check how to do this at the following link:
Let me know if the shared information has been helpful. Regards.