  Let $\Delta_\lambda$ be the Bessel operator on the upper half space
$\mathbb{R}_+^{n+1}$ with $n\geq 0$ and $\lambda>0$, and $R_{\lambda,j}$ be the
$j-$th Bessel Riesz transform, $j=1,\ldots,n+1$. We demonstrate that the
Schatten--Lorentz norm ($S^{p,q}$, $1<p<\infty$, $1\leq q\leq \infty$) of the
commutator $[b,R_{\lambda,j}]$ can be characterized in terms of the oscillation
space norm of the symbol $b$. In particular, for the case $p=q$, the Schatten
norm of $[b,R_{\lambda,j}]$ can be further characterized in terms of the Besov
norm of the symbol. Moreover, the critical index is also studied, which is
$p=n+1$, the lower dimension of the Bessel measure (but not the upper
dimension). Our approach relies on martingale and dyadic analysis, which
enables us to bypass the use of Fourier analysis effectively.
