Its strips of wood you're seeing there, dont know if they go all the way through, why would these be used? As for the wall construction I've not heard of that type, have you any info about it?
I was hoping the strips would be wood, it was difficult to see from the photograph. Closer inspection will reveal what they are used for. Before we started using various rawplug anchors, the traditional anchor was a built in timber pad. I strongly suspect these serve the same function. I would expect you to find the vertical studs nailed to them.
I have never seen that particular form of construction before, but it seems to be a form of dry lining to solid 9" walls. Perhaps your caviity wall surveyor was fooled by the apparent extra thickness to the wall construction it creates.
If your timber studs are dry and sound then you could use them again if you want to, adding a high performance insulation between the studs, before putting back the plasterboard. Add a vapour barrier over the insulation if the insulation does not come with any.
If you chose to use an insulated plasterboard, then it needs fixing as recommended by the manufacturer. I would need to check if you can dot and dab because of the insulation on the back!
Your suggestion to use an insulated board on the existing studs with added insulation between the studs, is in my opinion, probably the best way to go.
Edit: As for solid 9" walls, they will be fine if you have no signs of dampness penetrating. Thermally they are bad news but you are on course to make improvements. My own home is solid 9" brick and has no dampness. The same will be true of thousands of others. Occassionally a solid wall will allow penetrating dampness, but it is not the norm. English garden Wall bond is the way bricks are laid. A row of headers approx every six courses
Brick Bond types -->
http://en.wikipedia.org/wiki/Brickwork I stand corrected, I think yours would actually be "Scottish Bond" according to wikipedia (look under the heading of American Bond), you learn something new every day