Proving the language which consists of all strings in some language is the same length as some string in another language is regular
Remember (or come up with) the proof for

$\qquad \displaystyle L_1, L_2 \in \mathsf{REG} \implies L_1 \cap L_2 \in \mathsf{REG}$.

Do you see how to modify the proof for your setting?


   Abstracting the equality of lengths thing, come up with a construction for an automaton for

 $\qquad \displaystyle L_l = \{ w \in \Sigma^* \mid \exists x \in L.\, |x|=|w|\}$

 for given, arbitrary $L \in \mathsf{REG}$ over $\Sigma$.  


Do you see the connection?


   Now note that $L = A \cap B_l$.