Well to start with, i'd slacken the strings, pull them cleanly out of the grooves, and then reset and tune them again, it'll take you all of 3 minutes max to see if it behaves the same way again.
Hold up, what guitar is this on? Does it happen both fretted and open?
Obviously, the bridge shouldn't rattle, it would have rattled with the lighter gauge too, but since you've gone from lighter to higher gauge, and you're complaining about the Low E and g, i'd say straight away (i know you said it's the bridge that's rattling) that it would be fret buzz and you haven't set it up properly.