"If it transfers, the Lyric Opera doesn't have to produce it right?"
In theory, they wouldn't have to. That being said, it is their production. The costumes and set designs, etc. Are likely owned by them as the producers of this production, so their future involvement would still likely be present on some level.
If you are hoping for a transfer, I really wouldn't hold my breath if I were you. It isn't impossible, but it seems highly unlikely.