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Again, the thing is that the client has made the decision and he definitely wants to do this. I'm not planning to fight it any further as I've already explained to him the difficulties and he appreciates them and has made the decision to do it anyway. The new site won't be exactly the same, but it will have broadly the same set of products, albeit described in a very different way. The situation is undoubtedly difficult and undesirable, that's for sure. But the client has made a business decision and now we have to find the best way through it.

Absolutely. I definitely do not believe that Google is just ignoring bad links now, and Penguin can be a really hard one to debug once you think you have been affected especially since they aren't really (from what I have seen) announcing them anymore because they folded it into the main algorithm. When you have a manual penalty for manipulative links, sometimes you have to go the hard way of disavowing after you've tried rounds of getting them removed and it's still not enough. This may be the main reason, along with negative SEO, that Google has left that tool in Search Console. I don't expect it to go away anytime soon. Interesting times we live in.

Helen, I know people who have had success in reducing the number of links within mega menus by turning some of them (after the first two levels, for instance, but you could get much more sophisticated if you wanted) into javascript links. If the javascript is not too complex Google will still have no trouble getting to those pages, but the links won't be "hrefs" and therefore won't waste pagerank on pages that are not as important relative to the others. The upside to this is that the links are still there for users, assuming that is a good thing. As someone else mentioned, consider whether having those links there really is good for the users, or if they'd rather see a simpler menu. The search engines are beside the point in that case. Any time an eCommerce site experiences slow, steady traffic drops I always look into the uniqueness of their product copy. That is often a sign that they are sharing product copy with other sites, either due to manufacturer description use, or by publishing feeds to 3rd party sites like Amazon, eBay or price comparison shopping engines. Good luck!