The easiest solution for 90% of unexpected rPi power-down cases is a small uninterruptible power supply.
Laptops basically have a UPS built-in.
Doesn't the Pi 3 expect up to 2.5 amps? I imagine some of that is for powering USB ports, so you could probably get away with 1.5 amps if you're not using them, but even that's a lot for laptop USB. I'd be surprised if any offered more than 1 amp.
However, cheap is not how I would have described it ;-)
I did some research into this as well and I'd gladly consider this as an option but most power banks can't do this. And the ones that do don't tend to advertise it. Every time someone says they found one that can do this, in some forum or whatever, they are no longer available to buy.