Already done anyway. Read up on the AFIRS "Blue Box" from Flyht Aerospace.
During normal operations, AFIRS periodically sends out routine data (speed, position, fuel status, etc.) via satellite to enable real-time flight following.
But upon detecting any anomaly, AFIRS switches into "high resolution" mode and will start live-streaming blackbox data (starting from the previous 20 seconds prior to the anomaly). This data can be automatically routed to the company, XAA, TSB, etc., for real-time analysis and archival.
AFIRS is already certified and operational here in Canada with First Air since last year.