with a laptop, it's tricky because as you said, you will have to push the button. What I can recommend is to get a Raspberry Pi, and set that up with Fldigi. The Pi can be left powered on - it does not use much power - but it also powers up when you turn the power on via e.g. a smart plug. Just make sure that you shut it down before you turn the power off, otherwise the memory card can get corrupted, and it might not boot anymore.
