TLS was indeed closer than LYS, and BOD was even closer from the point they diverted.
Both TLS and BOD have hospitals nearby the airport while LFLL/LYS is kind of in the middle of nowhere.
Therefore I don't think it was a medical.
Only thing coming to mind is that LFLL has a 4000-m long runway. Would that be the reason for the choice of alternate?
Whatever the reason, it seems they wanted to put her down asap, +300kts below FL100 per FR24 ...