UPDATE:
I bought that 15v 3A supply in the first link and it doesn't work. The convertible powers on, the screen flashes, but then nothing happens. Occasionally it'll do a single beep, and once or twice I even got it to begin to seek the A drive, but it always shuts off almost immediately. I'm certain that it's the supply which is at fault, because the machine works perfectly using the car adaptor, so I can only assume it's tripping something in the power supply and causing it to shut off. I've now ordered a "car to wall" adaptor and will post an update about that when it arrives.

UPDATE:
The exact device I bought was this. After plugging in the IBM car supply into it, the 5140 works just as it does when I run it from my actual car. I can now verify that the machine functions exactly as it should with no issues.

Do you happen to have a multimeter?
I'd be curious to know exactly what voltage the car adapter is putting out.

A lot of older laptops used 19v supplies, and "car adapters" had to step the 12v
from the car to run the laptop. That might be the case here, but you would want to be
certain before putting any other supply on it.

Does the machine, the car adapter (or the wall adapter if you have a non-functioning one)
say what DC voltage is specified.

Is the the car adapter very large, does it have electronics? Step-up systems from that period tended to have sone "stuff" inside.

If you decide to go with the car-adapter on power-supply solution, be aware that a
car is typically 12v ... as they are subject to changes (when running etc.) a couple extra
volts is usually accounted for, but as a long term solution I would go with a 12v supply.

Btw, a supply capable of higher current is almost never a problem under typical loads, the
only time it can be is if the required current is VERY low - a supply designed to output significantly
higher might not regulate on very low loads... 2.7 <> 3 would not be a problem.