You can remove large areas of copper pour exactly how you have imagined. The most important tool you need is a specific shaped chisel. If you have access to a belt sander, you're set.
Take a 1/8" to 1/4" steel rod or allen wrench or what not, and grind it into the shape of a regular wood chisel. Roughly 25 to 30 degree bevel. Then very gently round the sole/back of this chisel into a slight arc (side to side, not front to back). You can give the back a little angle, here. It doesn't have to be parallel, I,e you don't have to grind the entire length of the rod, or anything. Just the tip. Then sharpen it.
You work this chisel under the copper with the same orientation as paring with a wood chisel: back down, bevel up. But you don't push it forward. You just constantly twist the chisel on the radiused back, which levers/pries the copper away from the board on either end of the sharp edge. Once you get it started, the copper will slowly but surely peel away with this gently prying. Once you get a large enough tab, you can just peel with pliers, but the chisel could do it alone. And the chisel will take care of odd parts the rip off and get left behind. Aside from the gouges you make at the initial attack point, you can do this leaving virtually no marks on the substrate, at all. Once the peel has started, you do not need any direct contact between the edge and the board... it is contacting only the copper.
Anyhow, this works for me, and it is very neat and easy. If you like to try it, I am pretty sure you will agree.