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.
This is the beauty of the web- the transition to a sane architecture can be done page by page- without any visibility to the user. Every time you have to make a change- fix an old feature or add a new one, you replace it with the new architecture. Even if it's not a whole page- you can load in a partial with javascript. The challenge is holding back from going too deep on the refactoring all at once. Functional tests around the system have to be added to keep your sanity as part of the change process.
This is the best response. To be a bit more general: evaluate whether the client is actually willing and able to take the sane course of action starting NOW and continuing over the long term. If not, and you stay, there's no advice that's going to be much help. Your life will be hell.
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.