Denote by $\tilde{\mathcal F}$ the Fock space representation of $\widehat{sl}_t$. In fact it is even a $\widehat{gl}_t$-module, and as such irreducible. Now \begin{equation}\widehat{gl}_t= \widehat{sl}_t\oplus Heis \Big/ (Z_{\widehat{sl}_t}-Z_{Heis}),\end{equation} where $Heis$ is the infinite dimensional Heisenberg, and the notation is meant to indicate that in the direct sum, one identifies the centres. This $Heis$-action on $\tilde{\mathcal F}$ is not the standard one, but is by a smaller subalgebra of the standard one (inside the standard Heisenberg acting on $\tilde{\mathcal F}$, one essentially takes only the generators with indices divisible by $t$). In terms of this decomposition, one has \begin{equation}\tilde{\mathcal F} = V(\omega_0)\otimes {\mathcal F},\end{equation} where $V(\omega_0)$ is the basic representation of $\widehat{sl}_t$, whereas ${\mathcal F}$ is the standard Fock space representation of $Heis$. Note they both have central charge one, so the action factors through the quotient taken above. Everything so far is explained in much greater detail in Tingley's Notes on Fock space (Section 3E) or Leclerc's Fock space representations... (2.2.8).

Your $P_t(x)$ is the character of $\tilde{\mathcal F}$. In terms of the decomposition above, we understand it as one factor of $P(x^J)$, and the rest.

Now we have to understand how the rest becomes the character of the basic representation $V(\omega_0)$ of $\widehat{sl}_t$. There are several known constructions of the basic rep; the relevant one here is the Frenkel-Kac construction. That says that, as a vector space at least, \begin{equation}V(\omega_0) \cong {\mathcal F}^{t-1}\otimes {\mathbb C}(Q),\end{equation} where $\mathcal F$ is good old Fock space once again, and $Q$ is the finite root lattice of type $A_{t-1}$. This basically gives the rest of the formula, once the relevant weights are computed explicitly: you have $t-1$ more copies of $P(x^J)$, and $K_t(x)$ comes from the root lattice. The character of the basic representation is written in the form needed here for example in Theorem 20.23(d) of Carter's book Lie algebras of finite and affine type.

Denote by $\tilde{\mathcal F}$ the Fock space representation of $\widehat{sl}_t$. In fact it is even a $\widehat{gl}_t$-module, and as such irreducible. Now \begin{equation}\widehat{gl}_t= \widehat{sl}_t\oplus Heis \Big/ (Z_{\widehat{sl}_t}-Z_{Heis}),\end{equation} where $Heis$ is the infinite dimensional Heisenberg, and the notation is meant to indicate that in the direct sum, one identifies the centres. This $Heis$-action on $\tilde{\mathcal F}$ is not the standard one, but is by a smaller subalgebra of the standard one (inside the standard Heisenberg acting on $\tilde{\mathcal F}$, one essentially takes only the generators with indices divisible by $t$). In terms of this decomposition, one has \begin{equation}\tilde{\mathcal F} = V(\omega_0)\otimes {\mathcal F},\end{equation} where $V(\omega_0)$ is the basic representation of $\widehat{sl}_t$, whereas ${\mathcal F}$ is the standard Fock space representation of $Heis$. Note they both have central charge one, so the action factors through the quotient taken above. Everything so far is explained in much greater detail in Tingley's Notes on Fock space (Section 3E) or Leclerc's Fock space representations... (2.2.8).

Your $P_t(x)$ is the character of $\tilde{\mathcal F}$. In terms of the decomposition above, we understand it as one factor of $P(x^J)$, and the rest.

Now we have to understand how the rest becomes the character of the basic representation $V(\omega_0)$ of $\widehat{sl}_t$. There are several known constructions of the basic rep; the relevant one here is the Frenkel-Kac construction. That says that, as a vector space at least, \begin{equation}V(\omega_0) \cong {\mathcal F}^{t-1}\otimes {\mathbb C}(Q),\end{equation} where $\mathcal F$ is good old Fock space once again, and $Q$ is the finite root lattice of type $A_{t-1}$. This basically gives the rest of the formula, once the relevant weights are computed explicitly: you have $t-1$ more copies of $P(x^J)$, and $K_t(x)$ comes from the root lattice. The character of the basic representation is written in the form needed here for example in Theorem 20.23(d) of Carter's book Lie algebras of finite and affine type.

ADDED LATER

I believe $K_t(x)$ itself admits no simple product expansion for $t>2$. Set $t=3$; it is easy to see that an appropriate specialisation of $K_t(x)$ is the one-variable function \begin{equation} L(q) = \sum_{m,n\in\mathbb Z} q^{m^2+mn+n^2}. \end{equation} As I learned from Garvan some years ago, this function $L(q)$ of a single variable $q$ has a zero inside the unit circle, at $q_0=-\exp(-\pi/\sqrt{3})$, by a remark hidden on top of p.697 in this paper (which uses the same notation, except that it also introduces $a(q)$ for $L(q)$). This prevents $L(q)$ from admitting a product expansion in terms of $(1-q^n)$'s raised to a polynomial exponent in $n$, since functions admitting such expansions have no zeroes inside the unique circle.