It appears to fit.
Going by the backup device for the Doctor, it appears that a very small device is required for storage. That fits with the Moriarty storage device, which also contained his girlfriend, who was also self aware. However, going with the latter example, an Xbox huge laptop is needed to actually run the holograms in their rubix cube size data storage device, but at an unknown speed.
If I were projecting this into the present Trek, then the device to run the Doctor would be maybe the size of a modern 13 inch laptop, and capable of real time use. If I wanted to be very conservative, I would say Moriarty's device is 1/10 real time, or less, but could be the size of a Mac Book Air.
Either way, you could have the Doctor walking around freely using something the size of Bone's tool bag, and it could hold a few tools, even if most of it had to be filled with computer.