I've seen VHF WSPR reflected off aircraft exhibit doppler shift, and even doppler modulation caused by the vapor-trail or exhaust plume vortices. But this was line-of sight stuff and the WSPR data as received and stored doesn't contain this level of data. It is possible to save the demodulated FSK WSPR carrier as an audio file, and that could show the doppler effects, but very few hams save the raw signal data. You would need to use HF WSPR for long-distance detection, and I don't think the aircraft would be large enough to provide a usable reflection or other path anomaly at HF wavelengths. Also, for direction-finding or position localization you would need precise timing along with the raw signal storage, and the WSPR system only requires timestamp accuracy on the order of 10 or 100 ms and few hams try to do better than that. 10 ms only gives you about 3000 km resolution -- a pretty wide range uncertainty.
The whole thing sounds very unlikely to me.