Mason Anthony has a nice flow. Mason isn't one I'd use myself, but it's a decent name.
As for Brynnlynn? Oh boy. Why not just Brynn or Lynn? I like both of those names separately. At least she won't have to share her name with a bunch of classmates like she would have had she been named Madison, but I still don't care for Brynnlynn. It reminds me of a story I heard about a couple that was having trouble choosing between Ariella and Bella, so they went with Ariellabella. Yipes.