P0721 and engine shuts down

[spannerrash]

As you have already said, the gearbox codes should not shut down the engine. This either suggests you have 2 separate faults with the engine shutting down not logging any codes. So only the gearbox codes are show. Or the fault is affecting 2 separate systems at the same time. Gearbox and engine.

Have you checked for water ingres in the ecu box? Both ecu and tcu are located in the same box. This has happened many times and can be quite a common problem.

Have you tried squirting easy start or some similar solution into the inlet whilst trying to start the car? May at least tell you if the engine stopping is fuel or electrical related.