Could this be used in place of or in addition to a pitot-static system?
In place of pitot-static system
No, because it measures different physical quantity.
Laser Doppler anemometer measures velocity of the stream¹. But pitot-static system does not. Pitot-static system measures dynamic pressure and expresses it as airspeed at which that dynamic pressure would occur at sea level on an ISO standard day².
Since lift and control effectiveness depend on dynamic pressure, that is exactly the quantity the pilot needs to see. Laser Doppler anemometer does not provide it, therefore it cannot replace pitot-static system.
In addition to pitot-static system
Anything can be used in addition to what we have. But the question is whether there is a reason to.
For flying the aircraft, true airspeed is not needed and for determining the wind speed by comparing true airspeed and ground speed the airspeed can be estimated well enough from indicated airspeed, pressure and temperature.
However a laser anemometer has the advantage of being able to measure the wind speed ahead of the aircraft. This could be used for load alleviation (the system needs to know about the gusts a bit in advance so it can move the control surfaces in advance) and it could perhaps be used for detecting wind shear, though I am not sure how feasible such anemometer measuring wind speed hundreds of metres ahead is.
¹ A.k.a. true airspeed.
² That quantity is called equivalent airspeed. That is intended quantity, but the measurement is subject to various errors. The raw output of the pitot tube is called indicated airspeed and the value corrected for positional errors (due to the air-stream impacting the pitot tube at an angle) is called calibrated airspeed.