Sounds as though you already know the site has a crappy link profile. Will knowing if this was an issue for penguins, hummingbirds, pandas, ring-tailed lemurs or any other adorable animal affect your course of action? Or do you already know basically what needs to be done?
Yes, it's nice to know, when knowing is possible. Sometimes, though, we just have to carry on without knowing for sure.
See, the "big" named updates aren't the only changes Google makes. In fact, most years they put through over 400 algorithm tweaks. That's more than one per day. Most of them are minor adjustments that only affect a handful of pages, so it's not worth making a big deal out of each and every one. (Besides, they'd quickly run out of animals if they tried to name them all.)
It's possible that your client's site was affected by one of these minor updates, so the rankings/traffic change may have nothing to do with the big named changes.
It's also possible it has nothing to do with anything Google did -- perhaps one of your competitors has started a new SEO campaign that's pushing their URLs up in the SERPs. SEO is a zero-sum game. If one URL goes up, another has to come down.
The point is: rankings go up and down all the time. Unless you can clearly tie a change to the timing of a big named update, you will usually have very little idea why. Even if you can tie the timing to a big named update, at best you can only say you think it was that update that caused the change. We can never be certain.
Do what you know needs to be done to improve the client's site and their link profile. Knowing the name of a specific update that might have caused the drop isn't going to give you any additional useful information in terms of fixing the problem.