Borel sets with almost equal sections

This is a corrected version of a Q posted yesterday.

Suppose that $B\ne\varnothing$ is a planar lightface $\varDelta^1_1$ set, such that all its vertical cross-sections $B_x$, $x\in\text{proj}\,B$, are equal modulo countable. Is there a linear lightface $\varDelta^1_1$ set (or weaker, lightface $\varSigma^1_1$ set) $X$, equal modulo countable to all vertical cross-sections of $B$?

Comment. Picking a suitable singleton $\{x_0\}\subset \text{proj}\,B$, we get a required set $X=B_{x_0}$ in $\varDelta^1_1(x_0)$, of course, yet this is short of the parameter-free lightface $\varSigma^1_1$.