The SSD is OK since it gets to the boot screen and starts OK on AC. If it runs for a long time on battery after it is started, then the battery should be OK. However, you might try to run a stress test, high power use, to verify that. If one of the cells has a high internal resistance, it could drop the voltage when a higher current is being draw.
If it starts in BIOS and safe mode, then something that starts at the logon screen is overloading the battery - but that likely points back to the battery. The only thing I can think of that might would be the fans and potentially dirt causing them to draw more current to start. But they should also start in safe mode, but maybe not. Is there anything plugged into the LT? If so unplug anything not needed.