How do I convert 6V DC to 12V DC in a relatively small form factor.
A boost converter is what you're looking for. The amperage will be the most important factor determining the size. For approx. 1A you can expect the circuit to be quite small, too. There are literally hundreds of different ICs that would probably do the job. 

You can build the boost converter using your own parts and design a pcb, or buy a small module that's ready to run out of the box. This random one I found after typing "step up regulator" might fit your size requirements, although I recommend designing a board yourself. Datasheets for the many switching regulators include example pcb designs made according to all of the design rules.

As for the rest of your project - assuming you want to draw out 1200mA from the 12V output, that means more than 2.5A from the AAA batteries! I'm not so sure they can handle that. Batteries of different manufacturers, age, and under different physical conditions such as temperature will vary in the maximum current they can supply. Perhaps lithium batteries can fare better.