OK, it sounds like we might have it figured out here. If that receiver was designed specifically to work with the little speakers and sub it came with, and won’t allow any sort of adjustments, I think Brent is right. Your mains (the Polks) aren’t getting a full range signal, or even close to it. They’re probably being cut off somewhere around 200Hz, which is why you’re not getting any mid or upper bass from them.
The sub amp may be crossing over at a lower frequency (I think it ranges from 50 - 150Hz or so if you have the LPF engaged), which leaves you with a big hole where no sound is allowed in either the sub or the mains. The sound you are missing would usually be filled in by your mains, but the receiver won’t let them play it in this case.
If you can defeat the crossover on the BASH, you may be able to get that sound back, and it will play through the sub. Not ideal, but it should at least give you a little bit “fuller” sound.
I think the only real problem is that the receiver has the fixed crossover that is too high to use with bigger speakers. Your Shiva should play fine, if it were getting the right signal fed to it (at least that’s my take on it so far).