I wanted to check back in to both thank everyone for their recommendations, and to let you know that we ended up using Content Insight's Content Analysis Tool. It doesn't follow each link to its endpoint or down a couple levels, but it does list all the internal inbound and outbound links on a given page, which is pretty much what my team needed.
Unlike some of the other products I played around with, (import.io, webscraper.io, etc.), it was super easy to figure out and set up. So. Easy. With other products, I kept having to go back and try different configurations to get a usable/useful content scrape, and I still wasn't able to pull all that I needed.