Working With Awful Websites

I’d like some feedback on working with awful websites. My smallest client has a website with numerous broken links, CSS errors, canonicalization issues, and more. The site is a first class mess. They’re on contract, and they are VERY concerned about an increase in traffic. My concern is that if I focus only on SEO, all the traffic that comes in will bounce and they’ll cancel me for not producing leads for their sales team. I’m glad to spend the time fixing these errors, but that gets into web dev, which is not what they’re paying me for.

I’m curious to know if anyone else has dealt with a similar situation, and how you handled it.

Thanks,
Chris