OP question answer: Yes, it is possible to much more accurately measure airspeed with Doppler lidar, than it is with a pitot /static system.
I have used particle scattering and Geiger mode lidar for velocity and flow measurements, particularly (no pun intended) when a tempo/spatial map of the flow is desired. In atmospheric air, there are always some particles!
Here is an example of recent work in creating a lidar based sensor for aircraft velocity measurements: https://hal.archives-ouvertes.fr/hal-01111306/document This device provides TAS, angle of slip and AOA.
Here is a device which utilizes Doppler velocimetry. While the article suggests that the technique used is not lidar, there are several lidar techniques which perform analogous functions. http://optics.org/news/5/12/35
Here is a BAE concept which has been demoed at airshows, and utilizes Doppler lidar processing, utilizing a UV laser. http://www.baesystems.com/en/article/bae-systems-develops-laser-airspeed-sensor-for-aircraft
Here is a Doppler sensor for air data patent, which is 6 years old. https://www.google.com/patents/US8434358
Quoting this press release: " Airbus Group has completed successful flight tests on a fiber-optic, eye-safe, laser-based sensor system that delivers accurate airspeed information in the three axis at low and even negative airspeed. This range of capability is not possible with pitot tubes, the longstanding industry standard for airspeed sensors." http://www.marketwired.com/press-release/airbus-group-completes-successful-flight-tests-of-laser-based-airspeed-sensor-system-1978428.htm
A pitot tube / static port is a rather simple and inexpensive device.
IMUs and GPS proposed solutions are misplaced and will have a phenomenal error budget and do not accurately measure airflow.
NASA promoted a Rayleigh scattering lidar device a couple of years ago (2015?), so there are new developments.
A couple of decades ago the Navy was promoting an ultrasound doppler device.
While there are many ways of replacing the functionality of a pitot / static airspeed sensor, all are more expensive than a pitot system. However, the most likely technology to emerge with a reasonable sensor cost will be a Doppler lidar variant.