Thank you for the details, I've looked through your configuration, and made a few changes, that should affect the performace and should fix the issue. I suspect your database might have been overloaded, which caused the problem. You can find all the tweaks on this documentation article as well: https://goo.gl/Rn6lG6
1. I've enabled and generated the index table engine. It's made exactly for cases like yours. It gives a huge performance boost, and reduces the database usage. More about index table here: https://goo.gl/ooNYPY
2. I've also enabled the Custom Ajax Handler.
I have also re-enabled the search in the dropdown_boxes.php file in the theme child directory, to see if the problem re-appears. It appears to be working without issues now. I have not made changes to any other files.
I hope this helps!Best,
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