Borel selector reduction

For every set $L$ of marked lines $\ell=((\theta,b),m)$, where $\theta,b\in\mathbb R^4$ and $m\in\mathbb R$, assume that $L$ is compact; every $\ell\in L$ satisfies $\|\theta\|=1$ and $\langle b,\theta\rangle=0$ (with no condition on $m$); every unit vector $\theta\in\mathbb R^4$ is the direction of at least one member of $L$; and the custom packing dimension of the unmarked carrier $\{(\theta,b):((\theta,b),m)\in L\text{ for some }m\}$ equals $3$. Here, for a set $S$ in a pseudometric space, this custom packing dimension is the infimum over $d\in[0,\infty]$ such that there are arbitrary sets $S_n$ indexed by $n\in\mathbb N$ with $S\subseteq\bigcup_n S_n$ and custom upper Minkowski dimension at most $d$ for every $S_n$; the custom upper Minkowski dimension of $T$ is the infimum over finite $e\in[0,\infty]$ for which there exists a finite $C\in[0,\infty]$, possibly $C=0$, such that, for all sufficiently small positive $r$, $N(T,r)\le C\,(\operatorname{ofReal} r)^{-e_{\mathbb R}}$, where $e_{\mathbb R}$ is the real value of $e$, exponentiation is extended-nonnegative-real exponentiation, and $N(T,r)$ is the infimum of the extended-nonnegative-real cardinalities of finite, possibly empty, sets of centers whose open radius-$r$ balls cover $T$. Then there exists a measurable set $S$ of marked lines such that $S\subseteq L$; for every unit vector $\theta\in\mathbb R^4$, there exists exactly one marked line $\ell$ satisfying both $\ell\in S$ and $\operatorname{direction}(\ell)=\theta$; the same custom packing dimension of the unmarked carrier $\{(\theta,b):((\theta,b),m)\in S\text{ for some }m\}$ equals $3$; and