Is there a unique way to construct the overall spatial wavefunction for identical particles?

Determinants and permanents are not the only way.

There is a general result for the permutation group that, if you want to construct a fully symmetric wavefunction by combining the spatial and spin parts, then both parts must have the same permutation symmetry (i.e. must transform by identical irreps). This applies also for multidimensional representation of the permutation group.

If you want to construct a fully antisymmetric wavefunction, you must combine a spatial and spin part of conjugate symmetries.

As an example, it is possible to construct a fully symmetric 4-particle state by combining spatial states and spin states in the 3-dimensional irrep ${3,1}$, and a fully antisymmetric state by combining a spatial part in ${2,1,1}$ and a spin part in ${3,1}$ (both of which of course are $3$-dimensional).

The explicit sums of products of the various parts are given by the Clebsch-Gordan coefficients for the permutation group (in my example, those would be CG’s for $S_4$).

This method does have the advantage of providing you with state of specific total spin but algebraic cost increases with $n$, and tables of CG’s for $S_n$ aren’t so common (much less pre-packaged codes).

On balance, determinants or permanents are more common, and chemists in particular have developed various clever methods to extract angular momentum or spin information when needed.

The fact is that all states of an N-particle system can be written in this way. This is because the Hilbert space of such a system is a subspace of the $n$-fold tensor product of the one-particle Hilbert space. However, there are certainly other methods of describing states in this Hilbert space. A particularly useful one is in terms of occupation number. Since the particles are indistinguishable, the state is completely determined by saying how many particles are in a given state. In the case of bosons this number is unrestricted. In the case of fermions this number is restricted to be $0$ or $1$ due to antisymmetry.

To be specific, consider a one particle Hilbert space $mathfrak{H}$ spanned by states ${|phi_1,phi_2,cdotsrangle}$. The correspondic bosonic/fermionic $N$ particle space is spanned by kets of the form $|n_1,n_2,cdotsrangle$ with $n_1+n_2+cdots=N$. It corresponds to $n_1$ particles being in the state $|phi_1rangle$, $n_2$ particles being in the state $|phi_2rangle$, etc... In the bosonic case $n_1,n_2,dots in{0,1,2,dots}$. In the fermionic case $n_1,n_2,dots in{0,1}$.