I'm thinking something simple like
Beatrix Anne or
Beatrix Rose is fitting simply because one does not hear
Beatrix all that often these days--at least where I live--and it would be something simple to pair with the more unusual
Beatrix. I don't know though, I'm not usually into "common" or "filler" middle names. But in this case they work.
Beatrix Margaret is also sweet, as is
Beatrix Claire.