So then I added the copper shielding to the cavities.
The pickup routes:
Attachment 4462
The control cavity and cover:
Attachment 4461
It took a bit more than one of the shielding tape packs from PBG, mostly because of the cover, but partly because I screwed up and wasted some when I started.
Next time I use this stuff I hope to be much neater.
Might get some soldering done tomorrow.