My Lavellan would of course, be initially shocked.
But she is very cynical of the Dalish, and being part of a clan like Lavellan doesn't help her detachment from them either. Through the course of the game, she slowly let go until at the final romance scene, she was eager to get her vallaslin taken away; albeit a bit hesitant at first only because she had lived with them for so long. She will feel it though, if she ever goes back to her clan (they're still alive) and is viewed as an outcast because she discarded her vallaslin. She'll learn and grow, she's pretty young!
I veered off track on that but anyway, she wouldn't really mind. She'd be fascinated to learn more about him and see the contrast between the Dalish tales and who he really is. She would also be a bit less soft edged and willing to argue, to learn. It wouldn't affect their relationship unless he were to do something pretty bad. She'd try to help him come up with solutions or better means of achieving things. Even if he were to do something wrong - which she'd surely acknowledge - she'd still love him and be by his side regardless. (If Hawke can do this with a romanced Anders, so can she with Solas!) If she sees that he veers off from his path of trying to restore order and down into a more sinister path, she'd try to stop him because she cares.