I don't see a way to remove individual inner pads. The "Restring" section of the design rules can control the size of the inner pads, including 0 for no pad, but it only does it for all of them.
The pads are created with a single round flashed aperture, so they're easy to delete with an editor, as you suggest. Even a simple editor, such as gerbv on linux, can do this. But of course you'd have to re-edit the gerbers every time you ran the cam processor.
I think I would do this by creating a part with all the layers manually defined. Any pin that shouldn't have an inner pad on one or more layers I would build using a drill and then place circles on the layers that should have pads. Likely the DRC won't like this, so be prepared to override the errors for this part. You would then also need to manually route to the pins after placing the part on the board. Or, if you have the space, you can create the part with through-hole or SMD pads away from the pins that would serve as connection points (and make the part work normally in the schematic editor).
I would also check with your PCB fab house to make sure they don't have an issue with copper missing on the inner layers when they plate the holes.