The driving transistor for the charging LED is integrated directly into the BMIC, which is a nice feature. That's why you can't finding a driving transistor. (By contrast, the testing LED on so-equipped calculators does have a driving transistor. Two, in fact, as it's a two-color LED.)
However, you do not need to use a donor calculator; you can just buy the BMIC directly off a site like Digikey. The chip is a TI LM3658 and is marked L155B. A voltage surge could definitely damage the BMIC, so if you've got the skills, replacing it is definitely a worth a try.
Another possibility is that the USB port's boost converter is stuck on, causing the calculator to try to charge itself, which will actually waste a fair bit of power. The boost converter supplies +5 V to the USB port when the calculator is in host mode. If you never need to use something like a TI Rover or CBL, you might be able to identify and disable the offending circuit, either by cutting the output trace, or by cutting the input to the boost circuit. Do note, however, that doing so may also cause trouble with calculator-to-calculator linking; if you do experience trouble with linking, just swap cable ends, as you should only have trouble when a type A plug is inserted.
Additionally, TheLastMillennial, if the LED is on even when no cable is inserted, replacing the battery won't help. The BMIC will not drive the charging LED when no external power is supplied; in fact, it shuts down completely.