I'm afraid you can't use both integrated and discrete graphics. The vast majority of current systems can't do it. The only current systems that can use both integrated and discrete graphics have motherboards with intel's z68 chipset. If you want four monitors and your 6570 can't support that many, your only option is to either buy a card that can run 4 monitors (a lot of the higher end AMD cards can do this, though 2 of your monitors would need to have DisplayPort output or would need to be connected to active DisplayPort adapters), or if your board permits it, get a second video card to drive the extra monitor(s).
Does the card not downclock on a single display setup, or only in dual display mode? Try a different driver revision. Sometimes certain driver revisions will mess up the triggers for when a card should downclock itself. It could also be that driving the 2 displays actually does push a weaker card like the 6570 a bit harder, so it won't downclock with the second display. The 6570 does not consume much power, or generate much heat, it wouldn't be too harmful to have it run at 3D clocks constantly. You could try using the AMD Overdrive section of the Catalyst Control Center to manually underclock the card if you want, though you won't be able to push it all the way down to 2D clocks using this method.