To be honest, a lot of those first names are very close to Ava. Maybe not full, but when they're nicknames, they're kind of similar.
Alice could be Allie, its not that close to Ava, but I can see it being slightly problematic.
Same with Annie.
Vivien as a whole isn't that close to Ava, but Vivi and Ava kind of makes me trip over all the V's.
Evelyn as a whole isn't too much like Ava, but Eve and Ava? Uh...
So I'd say my vote goes to Lily, not just because it looks good next to Ava and isn't problematic, but also that it sounds great with June.
Lily June & Ava!