When painting, how to move part of a design from a layer to another?

Select the area, Copy Merge, Paste in a new top layer that will cover the area in the lower layers. Then methe the layer withe the layer yoe want the content in.

You can also cut the pixels out of the lower layers. However, there can be problems when there are smart object layers, and vector shape layer involved.

If you just dealing with pixels layers cut, paste, merge.  However, cutting the pixels will leave a whole in the layer the pixels were cut from. The pixels you changed by accident will not be recovered.

To recover pixels you need to back up in histort then do the update the way you want it done.