@sarmcl, though I don't know the specifics here (e.g. is it a multi-location business, did the client also relocate, etc.), here are some SOPs that may help:
1. 80/20 rule. Focus on the 20-25 sites that matter in some discernible way. No need to bother with the Brownbooks of the world.
2. Of those, prioritize the sites that show up for brand-name searches or on page 1 for the terms you're trying to rank for. Rather than the data aggregators or the smaller, less-visible sites.
3. Don't try to get all the cleanup done at once. Do a 2nd round of work a month or two after the first, and maybe a 3rd round of work a couple of months later. That's the only way you'll take care of some of the stragglers.
4. If a site requires payment just to fix the listing, skip it. (If it's a site you'd want to be listed on anyway, then sure, it might be a good use of money.)
5. If the client is somewhat OCD about the listings, make sure he or she understands that some listings require owner-verification, and that your ability to work together on those listings determines when or whether they get fixed.
6. It's more important to have a listing on a site that IS correct than to be free of duplicates So if, for instance, you're having a devil of a time claiming a listing on a certain site, I'd probably first add a new listing and then try to get the duplicate removed later. Takes a little of the pressure off.
7. Don't expect much or any rankings bump from the cleanup. It's still worth doing, generally, but not worth chewing up your whole schedule and putting off the other work you plan to do.