We did have that kind of experience. While all of your links are low end, we had a major problem because some of their links were good.
So we came up with a 6 week strategy. Simultaneously we started a content campaign and vertical business partnerships with links to the site.
After two weeks the client company gave a month's notice period to his old SEO company.
Once the period was over we did fall in ranks gradually for a fortnight (that was panicky) but we also started ranking for some alternative keywords (hence the client's business did not dry out).
We regained most of the rankings after 6 -8 weeks.
The downs of this strategy was the client had to pay two companies for 6 weeks, Their businesses went down for three months; but close to a year they have made up for the cost.
My tips would be to 1)mentally prepare the client for the loss; you can add the ROI sheet to your proposal. 2) include some other keywords in your campaign so that you can gain rankings immediately and the client's business does not dry out.