I don't think it's related to the PHP configuration. The error message means, that the database server gets overloaded. It might not be able to handle the search requests.
Try turning off the “Trigger live search while typing” option: https://i.imgur.com/Y2u9uQP.png
It saves a lot of performance, as the plugin is only triggered when the user hits the return button or clicks the Magnifier.
Since you have a lot of traffic, I also recommend using a cache plugin, like WP Super cache (you have to enable the cache after activation). It saves a lot of performance, and lot's of database connections.Best,
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