I may be missing something, but it seems to me it should be possible to write software that would calculate air speed without pitot tubes (and without direct reference to ground speed). INS (or GPS) should be able to provide the airplane's longitudinal acceleration at any point in time. From that and the airplane's mass you could calculate the longitudinal force. Subtract the thrust you know the engines are producing, and you'll have the aerodynamic drag.
The variables needed to calculate drag are air density, drag coefficient, and velocity (airspeed). It should be possible to calculate the drag coefficient from the airplane's configuration and attitude, and the density from altimeter and OAT. Plug those into the formula with the known drag, and out should pop the airspeed.
Much easier done than said, I'm sure. But a simulator takes control inputs and a number of other parameters (including airspeed) and calculates (I assume) an acceleration from which it determines the airspeed in the next "frame." The software I'm envisioning would be similar, except that instead of starting with airspeed and calculating acceleration, it would start with acceleration and calculate airspeed.
While I'm on a roll, there may be a simpler way: monitor the effectiveness of control inputs. If the pilot or FBW commands a certain rudder deflection (for example), gyros and accelerometers should detect the resulting angular acceleration. From that and the airplane's polar moment of inertia, you should be able to calculate the force (sideways lift) generated by that deflection. That force would correspond with an airspeed.