Often, the only solution is to do 50-50 testing on CC to find what's broken. There are no mod checkers at all that will identify all broken CC. You can start with the Previously Broken mods and CC list (it's updated at every patch) to find ones to delete outright. After you delete, also delete the file called localthumbcache to clear out remaining in-game cached info (it doesn't usually matter, but when it does, it matters a lot). Instructions on 50-50 testing are in How to Use Mods and CC under vanilla and 50-50 testing . All these resources are pinned in this forum to make them easy to re-find.
If you narrow down a cause, please do let me know so I can keep the Broken CC list up-to-date.