It's not really about the evidence. Once they figure it out you can get them to play along a little longer, but they know.

The problem with doubling down as a habit is that the child's brain will mature more and eventually notice the lie. And then you did actual habitual betraying gaslighting instead of just the usual fairy tales to calm a child. Admit and acknowledge the child's mental development.
Parents start with having the full trust of their children. And there is quite the leeway before puberty hits. But trust once lost might never be fully regained. There is a real risk of creating serious trust issues.