Couldn't Mat be Gaidal Cain. At the end of a crown of swords before he is caught by the Seanchan he had a discussion with the seafolk. There he say something like being the swords master of master of blades (Sorry I've only have a Dutch copy so I can't give the exact quote). This would mean he is/will be the best with more than 1 sword and that Gaidal Cain). This would mean that Birgitte is the DotNM with can be. If you see in the books most of the main persons will marry someone that they see as just a fried or an ennemy (Egwene's dream of having Elayne's brother as her warder of being killed by him) Birgritte sees Mat just as a drinking buddy. And in her lives she could have had a relation with the Seanchan.

I know that there are a few failures in this theory. Birgitte could sence Cain when she was just ripped you of the dreamworld. She saw a kid but remember Mat is just an adult for 2 years and Birgitte has all her memories of past times where being an adult commes maybe on a later age.

She also sensed the direction, I have to search that passage to see where Cain was at that moment. Mat is maybe not ugly on the outside (at the inside he isn't quite that beautiful) and Birgitte stated that in the many lives that she had lived she had other looks sometime. Can anybody give feedback?