Security could actually be a way to sell the need for cleanup. Hire a team of independent auditors. If the code is in such a bad state as you claim, i guarantee they will find at least a dozen of XSS and XSRF issues, very likely some SQL injections and possible even a few RCE as root.

Maybe not the best way to increase direct revenue if the product is working, but it highlights the risk they are taking with such a shaky foundation, and puts the decision on managements table rather than yours.