$\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\operatorname{\mathsf P}}$

Let us denote by $I$ and $J$ your first and second displayed triple integrals, respectively. The innermost integral in $J$ does not exist in the Lebesgue sense. Understood as the "improper" integral, $\int_0^{\infty-}\cdots\,dx:=\lim_{b\to\infty}\int_0^b\cdots\,dx$, this innermost integral equals
\begin{equation*}
-f(t) g(u)\,\frac{\pi/2}{|t-u|}
\end{equation*}
for $t\ne u$. So, the value of $J$ would be
\begin{equation*}
-\int_{0}^\infty \int_0^{\infty} f(t) g(u)\,\frac{\pi/2}{|t-u|}\, dt\, du,
\end{equation*}
which would be $-\infty$ if e.g. the functions $f$ and $g$ are strictly positive and continuous. If $f$ and $g$ continuous and $f(t)g(t)>0>f(u)g(u)$ for some $t$ and $u$ in $(0,\infty)$, then no reasonable value can be assigned to $J$. So, I think nothing reasonably good can be done for $I$ and $J$.

However, things indeed work out well if, as you suggested, the minuses in the expressions of $I$ and $J$ are replaced by pluses. Then we can also relax your conditions on $f$ and $g$: instead of taking $f$ and $g$ to be "of rapid decrease which are $o(x^2)$ near zero", it will suffice to assume that $f$ and $g$ are in $L^1(0,\infty)$ and $|f(t)|+|g(t)|\ll t^p$ for some $p>-1/2$ and all $t\in(0,1)$. Instead of $I$ and $J$, here we are going to consider (with factors in the expression of the integrand conveniently rearranged)
\begin{equation*}
I^+:=\int_{0}^{\infty-} \int_0^{\infty} \int_0^{\infty} \frac{\sin(x(t+u))}{x(t+u)} f(t) g(u)\, dt\, du\, dx
=\lim_{b\to\infty}I^+_b \tag{1}
\end{equation*}
and
\begin{equation*}
J^+:=\int_{0}^\infty \int_0^{\infty} \int_0^{\infty-} \frac{\sin(x(t+u))}{x(t+u)} f(t) g(u)\, dx\, dt\, du=\lim_{b\to\infty}J^+_b, \tag{2}
\end{equation*}
where
\begin{equation*}
I^+_b:=\int_{0}^b \int_0^{\infty} \int_0^{\infty} \frac{\sin(x(t+u))}{x(t+u)} f(t) g(u)\, dt\, du\, dx
\end{equation*}
and
\begin{equation*}
J^+_b:=\int_0^{\infty} \int_0^{\infty} \int_{0}^b \frac{\sin(x(t+u))}{x(t+u)} f(t) g(u)\,dx\, dt\, du.
\end{equation*}
Namely, we are going to show that the "triple integrals" $I^+$ and $J^+$ are well defined (in the sense that the limits $\lim_{b\to\infty}I^+_b$ and $\lim_{b\to\infty}J^+_b$ exist), and we shall also show that $I^+=J^+$.

To begin doing this, recall that $f$ and $g$ are in $L^1(0,\infty)$ and $|\frac{\sin v}v|\le1$ for $v>0$. So, by Fubini's theorem, for all $b>0$
\begin{equation*}
I^+_b=J^+_b\in\R. \tag{3}
\end{equation*}

Moreover, $\int_0^b\frac{\sin xv}x\,dx=\int_0^{bv}\frac{\sin w}w\,dw$ is bounded uniformly in $(b,v)\in(0,\infty)^2$ and $\int_0^b\frac{\sin xv}x\,dx\to\frac\pi2$ as $b\to\infty$, for each real $v>0$. Furthermore, switching to the polar coordinates and using the that conditions $f$ and $g$ are in $L^1(0,\infty)$ and $|f(t)|+|g(t)|\ll t^p$ for some $p>-1/2$ and $t\in(0,1)$, we have
\begin{multline*}
\int_0^{\infty} \int_0^{\infty}\frac{|f(t) g(u)|}{t+u}\,dt\, du \\
\ll \int_0^{\infty} \int_0^{\infty}|f(t)|\,|g(u)|\,dt\, du
+\int_0^{2\pi} d\thh \int_0^1 \frac{r^{2p}}r\,r\,dr \ll1.
\end{multline*}
So, by the dominated convergence theorem,
\begin{equation*}
J^+_b\underset{b\to\infty}\longrightarrow\frac\pi2\,
\int_0^{\infty} \int_0^{\infty} \frac{f(t) g(u)}{t+u} dt\, du.
\end{equation*}
Thus, in view of (3), the limits in (1) and (2) do exist and are equal to each other, so that
\begin{equation*}
I^+=J^+=\frac\pi2\,
\int_0^{\infty} \int_0^{\infty} \frac{f(t) g(u)}{t+u} dt\, du.
\end{equation*}