Though Inmarsat analysis undoubtedly have been key to home in on a crash site, the Doppler method might not tell the full story. A jitter assessment would have to be performed on ping return clock rather than on return data. Maybe the sat guys here know how much raw data, or data about the raw data, is logged?
Normal telecoms (digital side of the sat modem at the earth station) would not yield anything useful. It is possible the raw analogue RF was being digitised and captured for diagnostic purposes. I guess that their business model sees them handle very short messages. If a message isn't received, they won't get a real-time complaint, but a complaint many hours or days later. It would be a useful function to be able to go back and tell the customer that their signal wasn't relayed because it was off-freq/garbled/too-weak/overmodulated/oh-it-really-was-our-fault/whatever.
Sat earth stations have very precise frequency and time standards. Incoming RF would be downcoverted to an Intermediate Frequency (IF) that would be supplied to the sat modem for demodulation. If this IF was digitised by a SDR receiver tied to the station frequency standards, then reconstruction of the original analogue waveform arriving at the earth station would be possible.
The analysis of this waveform has been the rocket science here. I would struggle to separate doppler shift from other sources of frequency drift, especially with so little data to work with. There would have had to have been some serious maths at work, coupled with data from other aircraft and detailed historical information about the exact beam coverage patterns of their sat.
Another possibility is that deeper analysis found some too-weak-to-demodulate-so-not-initially-noticed signal on the "IF-tapes" for their POR bird. The IOR plus POR recordings would then give triangulation.