That opens up other potential restrictions to getting this done quickly and easily. I wouldn't consider it best practices to create what is essentially a spam page full of internal links and Googlebot will likely not crawl all 4000 links if you have them all there. So now you'd be talking about maybe making 20 or so thin, spammy looking pages of 200+ internal links to hopefully fix the issue.
The quick, easy sounding options are not often the best option. Considering you're doing all of this in an attempt to fix issues that arose due to an algorithmic penalty, I'd suggest trying to follow best practices for making these changes. It might not be easy but it'll lessen your chances of having done a quick fix that might be the cause, or part of, a future penalty.
So if Fetch As won't work for you (considering lack of manpower to manually fetch 4000 pages), the sitemap.xml option might be the better choice for you.
