Hi Ed,
Google Analytics if fully GDPR compliant if there is a proper cookie policy disclaimer implemented – which is needed by default anyways, as WordPress itself uses session cookies, and that has to be declared.
You can actually access the full keywords list if you have a phpMyAdmin or similar database access tool. In the database table wp_ajaxsearchpro_statistics all the previous keywords are stored, with the number of searches.
Another possibility is to use Matomo analytics, whic can be integrated as well, here is a knowledge base article on how to do that.
I’m not sure if there is a deeper way to integrate with Clarity, as it’s main purpose is heatmaps of user behavior. Using it should already include the search itself, there is probably a way to get the phrases typed into the input field as well. Unfortunately I’m not familiar with how it works.