Pumping through the existing pump, could be possible. But it would seem to somewhat defeat the gains, of using the external pump. My thinking on this, would be that the vanes on the existing pump, would be a drag on the newly created, higher flow system. Eliminating the old pump, however, would create it's own problems.
Using both pumps simultaneously, seems like the most likely scenario. Don't know what kind of gains would be achieved though. It might help, or it might create unintended consequences. Like pushing the system pressure too high.
I think that the whole concept has potential though. If it can be done easily, then give it a try. It might work, and I don't think that it'll be a problem. I'd expect it to help the cooling system somewhat. My guess is, that it'll either help a small amount, or there'll be no improvement. If it could be used with an auxiliary radiator though...that'd be even better!
In the cooling system world. These problems are solved by a bigger radiator core, or a bigger cooling fan. Or as I suggested, an auxiliary radiator. Pumping faster, or more volume wouldn't achieve much more cooling power. Since the hot coolant isn't exposed to the radiator long enough. That's why you sometimes see those off road desert race trucks. With an extra radiator mounted in the bed.