Thank you for the details, I don't think SFTP is neccessary, at least not yet.
I'm assuming you are reffering to the shop search engine (instance 2, 'Product search'). I've looked up the back-end configuration, you have a redirection set, with no live search, but the redirection is forced to the default results page, not the WooCommerce results page. I have changed that to the woocommerce results page, now it will display the products there as expected: https://i.imgur.com/XObYAKA.png
About the performance
I believe the major bottleneck here is probably the high number of plugins. I'm not saying ajax search pro is not at fault, but having 54 active plugins definitely takes quite the time and resources to load. WordPress loads all the plugins, at every page view, and at every ajax request as well.
I have written an article about page performance in connection with plugins a while back: https://wp-dreams.com/articles/2014/03/maximum-performance-with-wordpress/
Although it's not neccessarly the number of plug-ins in every case, but the higher number of plugins, the higher chance of conflicts and negative performance impacts. It's a really hard thing to measure correclty. This also depends on your server performance and configuration as well.
I've made a few experiments on your site, disabled ajax search pro to test out the default search page loading time vs when ajax search pro is active, example:
– Default search (7.78s): https://youtu.be/nOym1HbQ2pM
– Ajax search pro (7.68s): https://youtu.be/L2DIGx6b4l4
Since there was page cache active, I had to add an imaginary query argument to the address bar to bypass that at start, please ignore that, it does not change anything.
Other measurments were very similar, except for some lag spikes, when either cases went above 15 seconds. I'm guessing that is because the server may have a thread limit, or it's close to it's performance cap.
I'm still not saying that Ajax Search Pro might cause an issue here, it's just the measurments indicate that something else might be an issue as well – as the tests reveal similar performances in both cases.Best,
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