Thank you!
The configuration seems to be just perfect, I’m afraid this is server/setup related. The performance tracker reports that the search process finishes within 0.29s on average, so the rest is take up by the whole wordpress application loading, theme, and all the other plugin loading time. Even when I enabled caching, the response time didn’t move, so it is definitely not the plugin causing the timeout. I can notice about the same delay on the loading of the back-end pages.
Turning off unused plugins, and overally reducing their number may free up some server resources, and potentially increase the peformance. Moving from shared hosts to VPS has also a great performance impact.