  We present numerical evidence of an interaction-driven quantum Hall plateau
transition between a $|C|>1$ Chern Insulator (CI) and a $\nu = 1/3$ Laughlin
state in the Harper-Hofstadter model. We study the model at flux densities
$p/q$, where the lowest Landau level (LLL) manifold comprises $p$ magnetic
sub-bands. For weak interactions, the model realises integer CIs corresponding
to filled sub-bands, while strongly interacting candidate states include
fractional quantum Hall (FQH) states at LLL filling fractions $\nu=r/t$. These
phases may compete at the same particle density when $p=t$. As a concrete
example, we numerically explore the physics at flux density $n_{\phi} = 3/11$,
where we show evidence that a direct transition occurs between a CI and a $\nu
= 1/3$ Laughlin state, which we characterise in terms of its critical,
topological and entanglement properties. We also show that strong interactions
generically stabilise a $\nu = 1/3$ Laughlin state even when the LLL is split
into multiple bands, and introduce a powerful methodology to extract its
topological entanglement entropy by exploiting the scaling of magnetic length
with $n_\phi$.
