Sperner family that maximizes the subsets of a partition
The correct upper bound is the sum of the $2^k$ most central binomial coefficients:
$$|\mathcal{F}_B|\leq\binom \ell {\lfloor (\ell - 2^k)/2+1\rfloor} + \dots + \binom \ell {\lfloor(\ell + 2^k)/2\rfloor},$$
or just $|\mathcal{F}_B|\leq 2^\ell$ if $2^k\geq \ell+1$. The sets $\{B\cap C\mid C\in\mathcal F\text{ and }A\cap C=A'\}$ are antichains. By the LYM inequality the union of $2^k$ antichains cannot be larger than the sum of the largest $2^k$ binomial coefficients. To achieve the bound, let $A=\{a_0,\dots,a_{k-1}\}$, and let
$$\mathcal F=\left\{C\subseteq A\cup B \mid \lfloor(\ell + 2^k)/2\rfloor- \sum_{i\colon a_i\in C} 2^i=|C\cap B| \right\}.$$