So the cold reality for this client is that the codebase will have to be replaced over time. You are not trying to escape legacy simply for the lure of something new, you are trying to escape insanity.
I would talk to the client about focusing your effort on helping them transition--a small piece at time--to a sane architecture. If they aren't open to that, they aren't your client. I've been a business leader in a position where we had to make really tough and painful decisions about coding projects gone awry. I don't envy their position, but continuing forward with this monster does not seem to be in the long-term interests of the company.