Private search not changing correctly to a user added search engine

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Description of the issue:
How can this issue be reproduced?

  1. Add new custom search engine in this instance I added kagi with my custom link to the search.

  2. I changed the default search and then clicked private search and clicked change to kagi.

  3. The system notifies that it changed it to Kagi my custom added search engine but keeps brave.

  4. I can change to other default options if I click change to startpage as an example and it changes it to startpage correctly

Expected result: It changes to the custom added search engine properly

Brave Version( check About Brave): Brave origin Brave 1.92.112 (Official Build) beta (64-bit)

Additional Information: picture of the changed to kagi but it keeping it as brave. I am using Artix linux.

Recently, another user wanted to change their search engine in private windows to Google (minus AI Overviews).

This appears to be the same issue. Let me ask @Mattches to emphasize the priority of these issues:

Left these in the issue as +1s and pinged a few people.