Part of the problem is you don't have but a fixed line level output on your integrated amp. That means that as you turn the volume knob on your integrated amp, the level of that output will not change, even if it is is of sufficient level to let another amp control full range of volume of the sub; you'll constantly be trying to level-match the speakers volume to your sub volume, which would suck. Pre-outs would have variable line level output, i.e. would change with volume knob of the integrated amp, thus maintaining a level match between speakers and sub as you change volume. Fixed line level output voltage tends to be lower than pre-out level, so there's the matter of achieving full use of the amp (by meeting its input voltage sensitivity requirements).
Only way I can think of to mickey mouse a setup like this is to use the second set of speaker terminals with a converter like I mentioned earlier....but really you just need a different combo of components that are intended to work well together.