I've done a LOT on the later HK system (since mine was supposed to have it and I found the amp was missing!) but I've gone the other way and fitted aftermarket as I didn't want to take the punt on a stupidly expensive amplifier!! Though one of my projects is working on a 'factory' style replacement to the DSP amplifier when they fail, so you can still keep the factory head unit (If only I'd thought of it 4 years ago!!)
But onto you problems more specifically... There are maybe a couple of possible options. 1 - there is a 'phone' connector in the boot somewhere which I believe was for the factory phone kit. There are audio input lines and a mute line in there - but how they interact with the system, I'm not 100% sure... IE I don't know if the audio comes through all the doors, or whether it just routes the phone in signal to the drivers door for example...
However, and external bluetooth unit might be able to be wired into there and get it working that way. Other options could include looking for a 'plug and play' kit that plugs into the head unit in place of the CD-changer, so that when you select CD, it then connects to the external input instead which would have your bluetooth device feeding into it... I think I've seen kits like this, but can't remember where at the moment. If I get some time over the weekend, I will look into it a bit more for you.
Also option 3 could be wiring in an external input to a 3.5mm jack connector (like is possible on the earlier Clarion head units) directly into the head unit - and then using that as an input to the stereo for a bluetooth unit.
I think that on the earlier head units, the external input that can be wired in overrides the radio/cd changer anyway - so if there's audio playing into it, then it blocks out radio/CD - and if there's nothing coming in then you can still use the other radio functions as normal. I am not 100% sure if there is a place in the Alpine head units to wire this kind of lead in, but it's something I've been meaning to look into at some stage aswell..
The final option would be to wire a telemute/bluetooth unit into the speaker wiring to the drivers door for example, so it would change over and route calls to the speakers in the vehicle - but this would have limitations on the bluetooth/music streaming side of things.
Let me have a look at the external adapters that plug in where the CD changer usually sits, as that's probably the best option - and I know I was looking at various things to do with that the other day, so hopefully I'll find a couple of links for you.
Hope this helps as a starting point...