ASI - derive anticipated airspeed (ie. What it would be if not acted upon by wind gusts) by calculating what airspeed should be at calculated attitude with known thrust setting at given altitude (temp you'll need a probe). Mach data by referring to temp and TAS (more about which to come!)
To adjust for immediate windspeeds, possibly use very accurate doppler radar which will reveal the speed of water particles in the air in the case of 'wet gusts' , and to measure the speed of dry gusts, the aircraft could issue a high pitched sound of a given frequency from the nose, receive it in the tail and wingtips, and using doppler effect calculations by comparing the frequency difference at reception could determine the speed of the air passing by the aircraft, like hearing a whistling train go by. I'm not sure how mach compressibility effects might upset this, perhaps someone here does.
I think you've truly lost it
If you presented this rather complex solution, with numerous inputs all liable to failure and error, I am sure the boffins would listen carefully... and then one would say "but we could just stick a tube into the air facing forwards?"
I'm really not sure why this pressure for a GPS derived "attitude"? Modern INS units are amazingly reliable, we have 3 of them, and even after internal failures, the attitude function is retained. Whereas any GPS derived solution is vulnerable to Uncle Sam just throwing the switch, and/or jamming? Can you think of a modern airliner accident caused by an attitude display fault? Since the STN 747 which was a mechanical instrument, and even then, the other 2 indicators were fine IIRC?
PS I think your Doppler thing will only work when the tail falls off
