I'm actually working on this exact tool, and I'll be releasing to BHW marketplace soon (within the next few weeks). We provide the infrastructure to check thousands of backlinks per day (which would fit your exact needs) and the software on top of it to display the results.
The only problem is... it's hard to check Cloudflare-protected websites. I mean, it's already compute/network-intensive to check millions of backlinks per day for every customer, but it's even harder to check Cloudflare because their entire business is about preventing scraping, bots, etc. We can get around it, but it becomes cost-prohibitive and just makes the prices way higher for customers to solve a problem that only affects 20% of websites. I don't know how much this matters to you, but if you want to be a beta tester/reviewer, give me a shout. Would love early feedback. Sorry to the mods in advance for namedropping something I'm working on outside the Beta Testing forum, but it solves OP's exact problem.