String break angle over the nut not sufficient?

[Gordon Bellerose]

Not knowing what guage of strings you are using, or how the nut has been cut, I am simply going from experience.

YES. You can use your nut files to file the slots on the bridge saddles. Actually, with a T.O.M. bridge that is how you set each string at the proper height.
The nut is still the most likely suspect in this scenario. If you have cut one of the slots at the same diameter as the string, you could be "choking" the string. What that means is that as the string vibrates it hits the side of the nut slot. If the nut slots are too deep, the same applies.
It could be also that the nut slot has a bit of a "slope". What I mean by that is the fingerboard edge could be a tiny bit lower than the headstock side, causing the string to vibrate against the lower portion.
Hope I've been helpful and not confusing.