Starting point for DC/AC-inverter
You can first transform your whatever value DC supply to a reasonable level in an isolated fashion. This could be 15-18V because 9 V AC would have around 13V peak voltage, so you want a little bit of headroom after that. This isolated DC-DC conversion can be achieved with multiple topologies. Search for isolated dc-dc. Flyback can be a starting point for that. There are many ICs and application notes that would allow you to achieve a pretty decent design in a reasonably short time. Then use a standard full-bridge+LC filter to generate the sine wave you want with the help of PIC and SPWM generation. The design complexity would be determined by your needs. If you want a low THD, stable sine wave then you would have to implement a closed-loop control, etc. However the basic principles and methods that apply to standalone 220V inverters would also apply in your case as well, with much easier voltage levels to work.