Hodge star operator on curvature?

It seems the resolution to OP's question lies in the difference between

1. the Levi-Civita symbol, which is not a tensor and whose values are only $0$ and $\pm 1$; and

2. the Levi-Civita tensor, whose definition differs from the Levi-Civita symbol by a factor of $\sqrt{|\det(g_{\mu\nu})|}$.