Ajax requests in customizer controls don't inject customized state as done in preview

[westonruter]

In the customizer preview, Ajax requests now get the customized state injected via jQuery.prefilter. In the customizer controls pane, however, this was not also implemented. This is important to ensure that that requests to the REST API will return back the expected previewed state.

Also relates closely to #38122.

[westonruter]

In 39138:

Customize: Ensure page/post stubs are included in listings and searches for available nav menu items.

Include the customized state in the Ajax requests to load items and search items.

See #38573.
Fixes #38122.

[westonruter]

I don't think this is important to address right now. It will play an important role in #37887. Otherwise, any plugins that do REST API calls can just use POST with _method=GET as shown in requests to fetch available nav menu items.

[westonruter]

• Use URL state for Ajax requests instead of wp.customize.settings. Move to base, but then can't use existing query params?
• Eliminate add request previewing wrapper so that underlying function can be overridden?

[celloexpressions]

Taking off of the no-replies list.