I think it's really beautiful. The pronounciation doesn't seem to difficult and I think people would get used to it, but the š might cause problems/not be used. I don't know how good English bureaucracy is at incorporating special characters? I really like Auszra too although I don't think that would help much with the pronounciation.
Auszra was the first Lithuanian national newspaper (I just looked it up). That's pretty cool.
Aušrinė is another similar looking name I really like. It's not in the database here but she is a Lithuanian goddess of the morning star. I don't know how often it's been used as a name though.