There are bluetooth satphones indeed but the Thuraya ones (
SG-2520 XT) are bug-ridden crap.
Iridium ones are a lot better but the operating cost is ~5x to 10x higher. On T I can fill my boots with tafs and metars for $0.99 (1 minute)
T offers 9.6k whereas Iridium is 2.4k.
I am aware of more obscure solutions using bluetooth-RS232 adaptors which may be another way but I am looking for something that will always work when you need it, which can't be said for bluetooth
I will already be connecting the unit to a bluetooth GPS (which is fed from a rooftop antenna) and IMHO having 2 bluetooth channels concurrently is pushing it