Alright, so we know that a mysterious singer is behind the events of Clockwork Lullaby and Full Moon Laboratory, right? I seriously think that Iriina Clockworker is that singer. Why? Let's look at the hints.
The first two hints come from Iriina's name. We know that Clockwork Lullaby series concerns Elluka, so why not Iriina, whose last name is also Clockworker? Now, let's look at Iriina's first name. At first glance, one would guess that it's related to Rin or Lily.
We know the singer of the songs mentioned above has the voice of Rin Kagamine, right? What if that singer is actually Iriina? Rin > Riin > Iriin > Iriina. We know Iriina is quite the inventor, so isn't there a possibility that she was making a new body for herself in Full Moon Laboratory? That new body could be based on the vocaloid Lily. Ririi > Irii > Iriina. Elluka could have stolen it from Iriina or performed a Body Swap with Iriina while she was in possession of that new body. This would explain Elluka's blonde-haired, blue-eyed form in Chrono Story.