I prefer
Elowen all the way. I don't immediately associate it with Tolkien, but that is the main reason why it appeals to me- it is similar to a Tolkien name without the associations that come with the characters' names themselves. I think it's beautiful.
It doesn't really need a nickname, but I guess both of those you've suggested would work.
Elowen Kate is pretty. I don't like
Elowen Violet as much, not for any particular reason.