Air taxi Captain reported airspeed management while hand flying a visual approach in turbulence resulted in an ATC low altitude alert and a CFTT event.

Narrative

We were inbound to ZZZ after have been turned off our previous approach due to surface winds exceeding our tailwind limits. After delay vectors we were advised the wind had swung out of the West/Southwest at 28 putting it within our xwind/tailwind limits. We were cleared for LOC in VMC with significant turbulence and windshear. When handed off to the tower we were recovering from significant windshear loss of airspeed which the autopilot and autothrottle were unable to keep up with. To deal with the loss of airspeed caused by the autopilot/autothrottle being unable to hold the segment altitude I disconnected the autopilot and autothrottle to more aggressively increase the power to recover our airspeed and lowered the nose to decrease the AOA. We had unrestricted visibility; no cloud cover and the airport in sight with plenty of terrain clearance. In visual conditions with adequate terrain clearance we would rather trade altitude for a quicker airspeed recovery. In our initial contact with the tower we requested a visual approach with an offset to the west which they approved after advising us of a low altitude alert. We then continued with the remainder of the visual approach and landing.