Problem:

Is it possible to have a taxonomy view with a nested post view and to be able to search the post view by keyword?

Solution:

Yes, it is possible, but you will need to disable AJAX search/pagination feature in both parent taxonomy view and child post view.

For example:

https://toolset.com/forums/topic/search-filter-for-nested-view-of-taxonomy-view/#post-1765893

Relevant Documentation:

Problem:
The user would like to refresh a view with AJAX.

Solution:
You can use the following function, but it will reload the whole page:

// replace 71 with your view ID
WPViews.view_pagination.trigger_pagination(71, 1)

I suggest you do the following:

Implement a search form in the view. The search will have at least, a search box and a submit button.
Configure the views to use AJAX for search results refresh.
Hide the search form with CSS.
Call the submit buttons for the views that you want to refresh:

jQuery('#my-submit-button').click();