On the other hand, it's hard for me to imagine Captain Smith thinking: "Yes, we've stuck an iceberg and we're going down by the head. But why even bother sending out an S.O.S. After all, we're just so darned far from our base. Also, we're the fastest ship out here--there's probably not an Olympic-class vessel near us, and who would come to our aid--a freighter? It's clearly just not worth the effort to fire up the Marconi."
That's why the time issue is the most plausible in explaining the final hours of Exeter's crew; the disease was killing the crew so fast, Dr. Carter (and staff) were overwhelmed, so the next action taken was call Tracey, then continue to try everything "on the shelf" to combat the disease.
By the time Carter realized it was a lost cause, he likely felt his first duty was to leave the recorded warning. Moreover, as CMO, he may not have the clearance or knowledge to launch a warning device, or had the time to do so.