Not sure if this helps, but if you turn on the Subject Field of Dublin Core in Premiere’s Metadata Display menu, it maps directly to keywords. If you update a keyword in Bridge, it will pop up in Premiere.
Surprisingly, it will even allow you to generate brand new Bridge keywords directly in Premiere. I’m kind of astonished that Adobe would hide away this ability (and even having labels that don’t correspond to one another) but it’s a great way to start generating meaningful metadata that can be shared across the Adobe product line.