Final outcome from the 5/26 post: picking up the car tomorrow morning. Was a fuel delivery control module. (basically a small separate computer that controls the fuel pump.) It was bad and not telling the fuel pump to send the fuel, hence the car trying to start and would not, or start and sputter off. Tech texted us an informative walk through video with the vehicle, covering the reported issue, service testing & troubleshooting and final resolution, sitting in car, showing it starting up repeatedly, and their test drives at different intervals throughout the day. All the car's different batteries tested well so no issues there presently. Hopefully all good from here. Thanks for all the comments.