I strongly suspect that the unexpected doppler shift is from jetwash.
That is, the principle source of noise from a jet aircraft isn't the engines directly (turbine spool), or the fuselage's passage through the air (turbulent white noise), but the stream of hugely-accelerated air which has exited the turbine(s) and is now shredding itself against the stationary surrounding air. The noise source therefor isn't a point (engine) but a linear source (the turbulent shred-wall interface between the jetwash and surrounding air), and it is moving rapidly backwards from the aircraft.
Which means that as the aircraft approaches you, the jetwash / shred turbulence is moving away from you, and is doppler-shifted toward lower frequencies, and once the aircraft passes minimum distance, the jetwash is streaming toward you, at a high fraction of the speed of sound, and should therefor be doppler-shifted upwards.
The insight that it was jetwash and not engines themselves making noise became clear to me when I lived near an airport with a road passing immediately behind the runway. I happened to be cycling past one day as a jet lined up for take-off, heading away from me. I was positioned directly behind it (and out of immediate reach of the jetwash). My first thought as the engines spooled up was "this is going to be loud" ... but it wasn't. Rather than the roar you'd hear when you were alongside the plane, all I heard was a loud spooling turbine whine ... until the jetwash roar itself returned to me echoed off mountains a few kilometers distant.
TL;DR: Jet engines don't make (much) noise, their exhaust does, and it has a markedly different velocity vector than the plane itself, or its engines, accounting for a different doppler signature.