Turning off heatbed for the last few layers. Good idea or not?
It's not a bad idea, and you should try it. But only on prints with some height, because:

The goal of the heated bed is to ensure adhesion for the first few layers. Without the heat on the bottom side of the layer, the layers above will pull those layers with it as they cool, causing the warp that you see. When your bed is warmer than the layers above, those first layers stay with the warmth. This continues as the layers above are pulled into adhering to the bottom layer instead of going rogue.

Print some objects that are taller than 10 layers, and see. When you succeed, get scientific and dial down the amount of layers until you see warp. Then you'll know which object height you can employ this.