Sorry about the delay. Had to pick up the kids from school.
I've never found, or heard of, a thorough way to do this with Bridge. Funnily, good ol' FastStone does it in a snap. If you don't have it installed, it's worth doing for this purpose. It's small, and fast to install.
Once you install it, you just navigate to your folder of jpegs that you're ready to upload, select them all, then go to Tools>Remove JPEG Metadata.
The only problem is that it even removes your copyright info. But since you've watermarked the images, I don't see that that's a problem.