You're right, the trackpad stops working when I hold the space bar in general, not just in Playscii. As for the picker, if I press the space bar and don't hold it, it disappears. Is there a way around that?
Yes, in the Tool menu select "Picker toggle key: hold" and it will change to "Picker toggle key: press". Now you can invoke and dismiss the picker by tapping space bar, which should free the trackpad up for what you need.
