Sounds a lot easier to reverse-engineer the Hand Controller Interface, especially since that comes with a handy RJ45 connector, which would make it cheap to build an adapter that allows you to poke around. Pretty certain that carries raw analog audio on any 2 of the 8 wires :), as they actually sell a mic adapter cable (OPC-589)! (turns out somebody did the work, and you can use that info) (oh furthermore turns out the ICOM manual actually has the pin assignments...)
What issues can I expect when using Bluetooth for packet radio?
Point is that the bluetooth headset profile uses a digital audio compression optimized heavily for speech – and not for preserving the digital audio symbols. It might work to transmit low-rate data through that, but it would be equivalent to adding a significant noise source to your system.
In other words: you'd be making your transmission worse without need.
To be honest, I'd expect a system that has up to 50W power output to be very cleanly modularized – you can probably easily identify the PA chain when you open it (it's what attaches to the antenna port and has big heat sinks!). Developing something that only generates / demodulates packet radio audio signal pre-distorted to counteract the Bluetooth compression as far as possible sounds a lot more complex than just building your own packet radio/FM modulator and directly piggy-backing that onto your radio :)
Hence, if I needed to build something that takes 1200bd of data for transmission wirelessly, I'd go for something like a Pi Zero, a RJ45 connector soldered onto a PCB that would plug directly on top of the Pi Zero and that contains both a 8V/5V voltage converter to power the Pi from the Mic port, and a I²S or USB sound card IC to generate/receive the audio, plus some kind of data connectivity (bluetooth, if you will), plugged in directly to the Pi. That way, I'd have something relatively small, which only transmits the raw data bits over your bluetooth link (no quality loss there), and does the modulation as close as possible to the radio. Of course, you'd need some kind of packet radio audio modem software running on the pi, but there should be something readily available :)