Does he have a strong reason why he is so against displaying his address?
Your conclusion about showing up in the local listings (7-pack, or what have you) is definitely right. Without an address on Google Maps and other citations confirming the address you don't really stand a chance. I think you just have to keep reminding the client that they are a local business and how important citations really are for local SEO.
You may still be able to rank organically (aside from the local listings), but again, if you are optimizing for geo-targeted keywords, it's going to be a challenge.
PPC can sometimes be beneficial in addition to SEO efforts, however, I wouldn't suggest putting all your time / money into it. We have clients in similar industries and the majority of the traffic is organic. This channel simply cannot be ignored.
Hope this info helps a bit!