I really like it.
I know what Briar means, and it's a known (if not, common) girls's name, but I would avoid Briar Rose for the Sleeping Beauty connection.
I love a lot of the simpler B names - Briony, Brooke, Blythe, etc. I think they're less frilly than a lot of the more well-used names these days, but still feminine.
I say go for it, but perhaps with a more syllabic middle name - Elizabeth, Persephone/Penelope, Aurelia, Felicity, etc.