[Android] Default search engine for private tabs does not update without closing all private tabs

[timchilds]

Description

When a user changes the default search engine for private tabs. The default search engine does not change until the user closes out all private tabs.

Review mentioning issue: https://appfigures.com/reviews/266342430751LVjJf0SHmdOIHzSlmlHw4YQ?lang=en

Steps to Reproduce

1. Open a private tab.
2. Search for something.
3. Change the default search engine for private tabs (Settings -> Search engines -> Private Tab).
4. Navigate back to existing private tab OR open a new private tab.
5. Search for something.

Actual result:

Search opens in old default search engine.

Expected result:

Search opens in new default search engine - the search engine selected in step 3 above.

Reproduces how often:

Easily reproduced

Brave version (brave://version info)

1.44.105

Version/Channel Information:

Current release

Video

IMG_8417.MOV

[Uni-verse]

Verified on Samsung Galaxy S21 & Samsung Galaxy Tab S7 on version:

Brave	1.47.165 Chromium: 109.0.5414.80 (Official Build) (64-bit) 
Revision	0f69b168d36a06cace4365e9f029fa987afa5633-refs/branch-heads/5414@{#1178}
OS	Android 12; Build/SP1A.210812.016

Test Plan:

1. Open Private Tab
2. Search something
3. Change SE in settings
4. Go back to existing tab and search something
5. Confirm search engine used was updated
6. Open new private tab without closing existing tabs
7. Search in the URL box again
8. Confirm search engine used was updated

• Verified change default search engine for private tabs is updated without closing all private tabs

Original SE settings	Search	Change SE	Search