Why doesn't the AC inverter support my laptop?

Most likely the initial current surge needed by the power adaptor exceeds the inverter limit so it shuts down.

Only solution is to get a better inverter.

In theory, it might be possible that the laptop power brick requires sine wave power, something that most cheap inverters do not provide.

Most cheap inverters emulate a sine wave output rather crudely: they vary the output voltage at +x, 0 and -x, where x is chosen to mimic voltages close to the peak of the sine wave. This is called "modified square wave". The "modified" means it spends some time at the zero crossing.

Some old inverters (not common anymore) might also emulate a sine wave output even more crudely: by varying the output voltage at +x and -x. This is called "square wave". I don't think you see these anymore.

However, my understanding is that most laptop power bricks are in general very tolerant of the power source and should work with modified square wave just fine. So most likely explanation is the inrush current, as Solar Mike observed.

Several solutions:

• If inrush current is the cause, find a smaller power brick
• If inrush current is the cause, find an inverter more tolerant of inrush current
• If modified square wave is the cause, find an inverter providing true sine wave output (note, they are expensive and have lower efficiency and higher standby current so I would try other approaches first)
• An entirely different approach: power the laptop directly from 12 volts via a car DC-DC laptop adapter with no inverter inbetween (not recommended as there can be high voltage spikes in the car electrical system, and the inverter has a better chance of absorbing them, whereas a direct 12 volt to laptop voltage DC-DC converter can pass them to your laptop, destroying the expensive laptop in the process)
• You can also find a smaller more portable less power hungry laptop. Mine has a 45 watt power brick and I have never found it to be insufficiently powerful for my uses. New laptops are in general extremely energy efficient, so they can do more with less.

The cigarette lighter plug can't power very large inverters: you might find a 150W inverter for the cigarette lighter plug. The cigarette lighter circuit usually has 10 or 15 ampere fuse (providing 120 and 180 watts, respectively). However, the fuses are tolerant of very short-duration current surges, so I'm sure if you find a good enough cigarette lighter plug inverter, you can power the laptop with it.

My opinion? If the existing inverter is fixed installation not using the cigarette lighter plug, try to use the existing inverter if at all possible, replacing the laptop or finding a smaller power brick instead. I would hate to mess with cigarette lighter plug equipment! The cigarette lighter plug is the worst connector for power delivery I have ever seen. Probably because it was designed for cigarette lighters and not general purpose power delivery.

Of course, a new fixed installation inverter could be an option. It's not impossible to replace an existing fixed installation inverter with a better one, just a bit more work than using a cigarette lighter plug inverter.