WordPress archives are assigned by WordPress itself, so this means an archive for taxonomies will be applied for all taxonomies, unless specified differently.
Toolset Views allows you this to some extent.
You can control each archive for each taxonomy separately.
But, you can not create Taxonomy Term archives, means an archive for different Terms.
With the WordPress API this is possible:
https://developer.wordpress.org/themes/template-files-section/taxonomy-templates/
You can create a template that is called taxonomy-{taxonomy}-{term}.php (replace the placeholders with your terms).
This will call this specific archive when taxonomy/term "xy" is called.
That template needs to be created with PHP, but in it you could call a Custom Search crafted with Toolset Views.
https://toolset.com/documentation/programmer-reference/views-api/#render_view
This is although not what we intend to do with Toolset, we document it only to the "Taxonomy" level.
Now, you could theoretically also listen to the current archive within some Custom ShortCode and use that, to conditionally hide or show entire HTML chunks of your custom search (using HTML conditional).
But I do not recommend this, as it might produce unexpected search results when certain handles are hidden.
On the hand, you can as well use Toolset Views settings - eventually they help to achieve the goal.
In the Archives (or Views), you have the Custom Search Settings, where you can choose what to display of a search, depending on what you already searched for.
So for example, if a certain search A) does return a set of results that has no field "xy" saved, the Search for that field will not be shown in the View. (see screenshot)
These are the possibilities you have with Toolset, please let me know if you need more help with it!