Bryne didn't even know about Gaebril being Rahvin, but he tells Gawyn in the end: "If you don't trust me, ask your sister. See what she says of it." And Gawyn accepts that advice with a nod.

Egwene and Elayne have firsthand informations on what really happened in Andor concerning Rahvin, Morgase & Rand, while all Gawyn heard before were mere rumors from a peddler. Therefore it simply makes no sense that Gawyn won't talk with Egwene and/or Elayne about this topic, and that he wouldn't rather believe them then that Rand isn't guilty, especially since Bryne argued basically the same AND considering that Rand is crucial for humankind's survival (except if Gawyn is under some kind of Compulsion, but that seems quite loony as well...).