I like 'The Love Project' or the 'The Love Projector'
Googling LV Project throws up loads of Latvia and Vegas stuff - probably not ideal. At least 'The Lyveden Project' takes you straight to us if you have an idea how to type it into google.
Googling 'The Set-Up' gets lots of entries on the movie and bands already using that name - if, at least, ones hardly likely to get that successful.
So something as unique searchwise as The Lyveden Project but easier to spell and less likely to put people off as some seem to be by that name.
Your example of the pronunciation -- Liv Dunn -- looks nice. (Reminiscent of Liv Tyler, I guess.)