I would seriously doubt that you can find an all-in-one integrated solution for what you want. You can still do what you want, it just takes a few different items. I do something similar. I have the HTC official car dock (which is great, but doesn't accomodate a case). When I drop my phone in the dock, it automatically turns on bluetooth. I have separately a bluetooth receiver that is plugged into an FM transmitter. The whole bluetooth/FM assembly sits in my center console compartment, where it has power and is entirely out of the way and hidden from view. I don't do hands-free calling with this assembly, because that would require the bluetooth part be exposed so the mic would work, but you could easily do that. Unfortunately, by default, you can't route the phone audio though bluetooth without using the bluetooth mic. There are some apps that allow more customized audio routing that might enable this, but I I'm not 100% sure about that. A quick search on amazon for "bluetooth fm transmitter" produces a number of products to handle this part of the functionality, then you just have to pick whatever dock you want separately.
Good luck.