You correctly say that such a state won't be unique. Indeed, for example, pick the 50%-50% statistical mixture of two random pure basis states in ${\mathbb C}^n\otimes {\mathbb C}^n$. Each of these two states may be entangled with one of two orthogonal states in the third ${\mathbb C}^n$; however, what these two states in the third factor Hilbert space happen to be is completely undetermined.
So even if you had a constructive method to find the pure state in the three-part Hilbert space, it would fail to generate unique results.
However, there is another, much more serious problem with your proposal: in almost all cases, it has no solutions at all. Indeed, it's easy to demonstrate this fact by a simple counting of degrees of freedom. Pure states in ${\mathbb C}^n\otimes {\mathbb C}^n$ are specified by $n^2$ different complex numbers (one of them, the overall complex normalization, is unphysical).
Similarly, a general density matrix on this space is a $n^2\times n^2$ Hermitian matrix so it contains $n^4$ independent real parameters (one of them is the trace which should probably be set to one). However, that's larger than $2n^3$ number of real parameters coming from $n^3$ complex parameters of a wave function in ${\mathbb C}^n\otimes {\mathbb C}^n\otimes{\mathbb C}^n.$ At least for $n\gt 2$, it's larger. So up to a measure-zero subset of cases, you won't be able to find any pure state that reduces to the given mixed state. The diversity of the required results (density matrices) is much larger than the diversity of the ingredients (pure three-block states) that you may use to produce the desired outcome.
Of course, if you only had a density matrix for one of the three blocks, and not two of them, you would be able to solve it. At least the counting of the parameters wouldn't make the existence of a solution for a generic density matrix impossible.