My interpretation of 'fair use' is rooted primarily in what I've learned in my Mass Media Law class this semester, so my input should be taken with a grain of salt -- just a full disclaimer here.
Uploading the clips without any sort of creative input from yourself (i.e. appearing on camera introducing the clips and commentating on them, etc) would probably not fall under fair use criteria. If you were to provide commentary, appear on camera, etc., then you might have a case for claiming fair use.
Unless you are a big YouTuber who regularly claims earnings from Adsense, then I wouldn't worry about Fremantle claiming the clips if you just want to straight upload them. What I said a few posts above wasn't meant to discourage uploading Price
clips to YouTube. I've done it. Many others here have done it. Heck, we even have an entire section of the site dedicated to clips and episodes that have been posted to YouTube!
I won't say there is 0% chance anything will happen to you, but I will say there is a <99.99% chance nothing will happen to you based on precedent. Fremantle will detect your video, claim it to gather what minuscule amount of Adsense revenue there is to make off it, and that will be the extent of their involvement. At the very worst, you might get a copyright strike from YouTube, but given Fremantle's generosity (or really, just their desire to profit off the clips we upload), that seems highly unlikely. The chances of any further action on Fremantle's part beyond that are even more minuscule.
In other words, you're good.