I think that the question has its exact analog in Classical Electrodynamics. Consider that in an inertial system $\,S\,$ we have a system of particles with
\begin{align}
n\left(\mathbf x,t\right) & \boldsymbol{=}\texttt{particle number volume density at } \mathbf x,t
\tag{01a}\label{01a}\\
&\text{moving with 3-velocity}
\nonumber\\
\mathbf u\left(\mathbf x,t\right) & \boldsymbol{=}\texttt{velocity of particles at } \mathbf x,t
\tag{01b}\label{01b}
\end{align}
Suppose now that we $''$charge$''$ each particle with the unit quantum charge $\,\texttt{e}\,$ of the electron. Then we have the following quantities of Classical Electrodynamics (omitting for convenience the space-time dependence)
\begin{align}
\varrho & \boldsymbol{=}\texttt{e}\, n\boldsymbol{=}\texttt{electric charge volume density}
\tag{02a}\label{02a}\\
\boldsymbol{j} & \boldsymbol{=}\varrho\,\mathbf u\boldsymbol{=}\texttt{e}\, n\,\mathbf u\boldsymbol{=}\texttt{electric charge current density}
\tag{02b}\label{02b}
\end{align}
If particles are neither created nor destroyed then in the electrodynamics analog the electric charge obeys the easily proved conservation law
\begin{equation}
\dfrac{\partial \varrho}{\partial t}\boldsymbol{+}\boldsymbol{\nabla\cdot}\boldsymbol{j} \boldsymbol{=}0
\tag{03}\label{03}
\end{equation}
In electrodynamics we define the 4-dimensional electric charge current density
$\,\mathbf J\,$
\begin{equation}
\mathbf J \stackrel{\texttt{def}}{\boldsymbol{=\!=}}\left(\varrho\,c,\boldsymbol{j}\right)\boldsymbol{=}\varrho\left(c,\mathbf u\right)
\tag{04}\label{04}
\end{equation}
so that the conservation law is expressed in terms of the 4-divergence
\begin{equation}
\boldsymbol{\square\!\!\!\!\square\,\cdot} \mathbf J \boldsymbol{=}\partial_{\mu}\mathrm J^{\mu}\boldsymbol{=}\partial^{\mu}\mathrm J_{\mu}\boldsymbol{=}0
\tag{05}\label{05}
\end{equation}
By analogy we define the 4-dimensional particle flux $\,\mathbf N\,$
\begin{equation}
\mathbf N \boldsymbol{=}\dfrac{\mathbf J }{\texttt{e}}\boldsymbol{=}\dfrac{\left(\varrho\,c,\boldsymbol{j}\right)}{\texttt{e}}\boldsymbol{=}n\left(c,\mathbf u\right)\boldsymbol{=}\left(n\,c,\boldsymbol{f}\right)
\tag{06}\label{06}
\end{equation}
where
\begin{equation}
\boldsymbol{f}\boldsymbol{=}n\,\mathbf u\boldsymbol{=}\texttt{the 3-dimensional particle flux}
\tag{07}\label{07}
\end{equation}
The particle flux obeys the particle number conservation law corresponding to equation \eqref{03}
\begin{equation}
\dfrac{\partial n}{\partial t}\boldsymbol{+}\boldsymbol{\nabla\cdot}\boldsymbol{f} \boldsymbol{=}0
\tag{08}\label{08}
\end{equation}
or its 4-dimensional version corresponding to equation \eqref{05}
\begin{equation}
\boldsymbol{\square\!\!\!\!\square\,\cdot} \mathbf N \boldsymbol{=}\partial_{\mu}\mathrm N^{\mu}\boldsymbol{=}\partial^{\mu}\mathrm N_{\mu}\boldsymbol{=}0
\tag{09}\label{09}
\end{equation}
Now, in electrodynamics it has been proved that the 4-dimensional electric charge current density $\,\mathbf J\,$ of equation \eqref{04} is a Lorentz 4-vector expressed also as the Lorentz 4-velocity $\mathbf U\boldsymbol{=}\gamma_{\rm u}\left(c,\mathbf u\right)$ times the invariant scalar rest charge density $\varrho_{0}\boldsymbol{=}\varrho/\gamma_{\rm u}$
\begin{equation}
\mathbf J \boldsymbol{=}\varrho_{0}\mathbf U
\tag{10}\label{10}
\end{equation}
By analogy the 4-dimensional particle flux $\,\mathbf N\,$ is a Lorentz 4-vector expressed also as the Lorentz 4-velocity $\mathbf U\boldsymbol{=}\gamma_{\rm u}\left(c,\mathbf u\right)$ times the invariant scalar rest particle number $n_{0}\boldsymbol{=}n/\gamma_{\rm u}$
\begin{equation}
\mathbf N \boldsymbol{=}n_{0}\mathbf U
\tag{11}\label{11}
\end{equation}
So the proof that $\,\mathbf N\,$ is a Lorentz 4-vector is essentially the proof that the 4-dimensional electric charge current density $\,\mathbf J\,$ of equation \eqref{04} is a Lorentz 4-vector.
Proofs that $\,\mathbf J\,$ is transformed as a Lorentz 4-vector based upon the conservation law \eqref{03} are false. That electric charge is constant in an inertial system doesn't provide any information about how it is transformed between inertial systems. It's a confusion between what is a constant (it concerns what happens in a system) and what is an invariant (it concerns what happens between two systems).
For an elegant proof by L.D.Landau and E.M.Lifshitz see my $\color{blue}{\textbf{ANSWER A}}$ here How do we prove that the 4-current jμ transforms like xμ under Lorentz transformation?. The proof is based on one hand upon the fact that the charge on a particle is, from its very definition, an invariant quantity and on the other hand upon the fact that the 4-dimensional infinitesimal $''$volume$''$ $\mathrm dV=\mathrm dx^0\mathrm dx^1\mathrm dx^2\mathrm dx^3$ is a Lorentz invariant scalar. A proof of the latter is given in footnote (1) of the aforementioned answer.
