On the 156 that I have, the speaker grilles simply prise out and the speaker can be changed without taking the door panel off.
With the Bose system, speakers aren't where you might think they are. There are no tweeters behind the 'Bose' grilles on the front doors. Despite apparent speaker covers on the rear shelf, the rear speakers are actually in the rear doors. The sub is built into the rear shelf and doesn't impinge on boot space.
Note that it's a 'sound quality' subwoofer. I used to have 15" subs in big carpeted boxes in the boot of my Lancia Thema, which drew some odd looks and made a horrible booming and droning noise, truly excellent. Then I had a 12" sub in a smaller box in my 164, which the new owner (in his 50s) immediately removed. Then I had a 10" sub in a round box I made specially to fit the spare wheel well of my Uno Turbo. That seemed to have a more pleasingly subtle effect (my friends thought it wasn't working) but I eventually removed it to save weight.
My last car had a 6" sub (FIAT Punto Sporting) and the one in the 156 is probably a similar size, I haven't even looked
Basically, I've grown up.
If you're also into the idea of a subwoofer as a sound quality improver (rather than as a loud boom on its own) then I suggest the 'active sub' sold by Jaycar - it's a flat box suitable for installation under a seat - includes a small driver (8"?) and suitable amplifier, all for only $150 or so, very convenient and practical (I haven't had to buy one yet but would do).
I actually find that with a good pair of 6.5s mounted well in the front doors, there's not much need for a subwoofer anyway.
To get the Bose system (which I think is a nice setup) you would have to transfer all the components and wiring - a lot of work. I think you're better to create your own setup, particularly as you already have an aftermarket head unit.
In the process you might like to soundproof the boot (deaden the panels, add some felt to the bulkhead behind the seat) and add sound deadening sheets to close holes in the sills, to generally reduce the road noise a bit. Expensive new tyres help, too, and if you can stay away from 17" wheels, so much the better (I failed to resist the temptation; obviously I haven't grown up enough yet).
I bought a sound level meter to prove that my 156 is still a lot noisier than my old 164s. In the 164, rolling on 15" wheels with sound levels only 68dBA at 100km/h, I've successfully made speakerphone calls on my iPhone (without carkit). My 156 recorded a 74dBA level at 100km/h. dB being a logarithmic scale, that's a lot louder. It's annoying because there appears to be lots of sound deadening in the floor of the 156. Maybe boxing in the doors (with sound deadening sheets) will help. I'm not sure it will ever be quite the solid luxury of an old 164 (and you can buy a 164 for about a grand these days...)