I was able to sort this out.

The "PS3 Controller" driver in the driver packages causes a conflict with the Atari VCS Classic Controller over Bluetooth. Uninstalling it allows the VCS Classic to work.

Obviously, this makes it so the only way you can use a PS3 controller wirelessly is to add it using the native driver, which doesn't provide rumble/force feedback. But the controller still operates file as a gamepad.

Not ideal, but I'd rather have both controllers available, instead of rumble.