Here is an attempt, somewhat rough around the edges but I think it works:

Claim: The range of the Fourier transform $\mathcal{F}:L^1(\mathbb
R)\to C_0(\mathbb R)$ is a Borel set in $C_0(\mathbb R)$ of the form

\begin{equation}
\bigcap_{k=1}^\infty \bigcup_{N=1}^\infty \bigcap_{m,n\geq N} E_{m,n,k}
\end{equation}

where each $E_{m,n,k}$ is an $F_\sigma$.

Proof: Consider the cutoff functions $\{e^{-a\pi|t|}\}$ and fix a
sequence $a_n\to 0$. It is a fact that a function $g\in C_0(\mathbb
R)$ is the Fourier transform of some $f\in L^1$ if and only if the
sequence
\begin{equation}
T_n(g)(x) := \int_{\mathbb R} e^{-a_n\pi |t|} g(t)e^{2\pi itx} dt
\end{equation}
is Cauchy in $L^1$, in which case if we put $f=\lim T_ng$ then $g
=\widehat{f}$. Let $R$ denote the range of the Fourier transform in
$C_0$. If we define

\begin{equation}
E_{n,m,k} = \lbrace g\in C_0: T_ng, T_mg\in L^1 ,{||T_n(g)-T_m(g)||}_1 \leq\frac{1}{k} \rbrace
\end{equation}

then $R$ has the claimed form.

To prove that $E_{m,n,k}$ is an $F_\sigma$, put $T_n(g)-T_m(g):=T_{mn}(g)$, explicitly

\begin{equation}
T_{mn}g(x):= T_ng(x)-T_mg(x) =\int_{\mathbb R} h_{mn}(x,t)g(t) dt
\end{equation}

where we define

\begin{equation}
h_{mn}(x,t) = (e^{-a_m\pi|t|} - e^{-a_n\pi|t|} )e^{2\pi itx}
\end{equation}

Note that by monotone convergence, $g\in E_{m,n,k}$ if and only if two conditions are satisfied: first, for fixed $n$, we need $T_ng\in L^1$. By monotone convergence this is equivalent to: There exists an integer $N$ such that for all integers $d\geq 1$,

\begin{equation}
|| {\bf 1}_{[-d,d]}(x)T_ng(x)||_1 \leq N.
\end{equation}

By dominated convergence the set of all such $g$ obeying this for fixed $N,n,d$ is closed, so the set that obeys this for some $N$ and all $d$ (with $n$ held fixed) is an $F_\sigma$.
Thus for fixed $m,n$ the set of $g$ for which $T_ng, T_mg\in L^1$ is an $F_\sigma$.
Additionally, for all integers $d\geq 1$, we need the condition

\begin{equation}
|| {\bf 1}\_{[-d,d]}(x)T_{mn}g(x)||\_1 \leq \frac{1}{k}.
\end{equation}

But similarly, the set of $g$ obeying this for fixed $m,n,k,d$ is closed in $C_0(\mathbb R)$, and we conclude that each $E_{m,n,k}$ is an $F_\sigma$.