I have been thinking of this. There were hourly pings which can be used to calculate the arcs. Distance between the arcs gives the average hourly speed towards the satellite. Doppler data would give the momentary speed towards the satellite at the moment of the ping. Doppler data would not give too much additional information, but would curtail the set of possible tracks.
I have always been sceptical that the logs would contain detailed enough data to estimate speed. Why would a desperate-for-storage-space engineer add logs for such extraneous information? (If she did, hats off as it may have made a difference to this investigation.)
Another possibility could be, that inmarsat had ping data from previous flights in their logs. If this data contains signal strength it could be feasible to calculate the radiation pattern of the a/c antenna. If there was a significant asymmetry in the a/c antenna it could be deduced whether the starboard or port side of the antenna pointed to the satellite.
Again, I doubt they would give scarce log space for signal strength data, unless it serves other purposes. I wonder if they haven't been using the slight variations above / below the equator that the satellite travels, with an assumption of constant speed and bearing, to eliminate the northern arc.