The camera already has a SD card slot, so I was hoping to somehow read from it as well.
Okay, well how? First you'll need physical access to it, won't you? So, start by tearing your camera apart, see where you can solder wires, and go from there.
I'm looking at this bluetooth module
Okay well, that's a serial device. Bluetooth isn't a generic protocol. All devices conform to a profile. Serial profile doesn't support file transfers. So you'll have to write some software on the PC side to communicate with your arduino and reconstruct whatever data you send into a "file."