Is there a way to catch copyrighted material that doesn't require us to review every video uploaded by our content creators?

Unfortunately there is no replacement for human content review. Social networks like YouTube might detect select copyright materials (but by far not all) according to their content policies. 

Lightcast.com being a hosting and distribution platform does not have such content policies, providing software and server space to store and distribute contents, while Lightcast.com's customers are in charge of publishing and unpublishing contents. 

If you accept contents from third-party producers, you have to make sure you know the source, the producer and their contents. This is, unfortunately, not something we can get involved in. There might be tools out there which could be downloaded and installed on your machines from where you initiated uploads, which scan video files for specific contents in question, prior to uploading them.