Thanks to P.Windridge's comment, I can now answer my own question.

Indeed the convergence to standard normal in question can follow from a triangular array version of CLT called the Lindeberg-Feller CLT. Proof can be found on Durrett's Probability: Theory and Examples (freely available online).

I reference the statement of the theorem from Durrett:

To arrive at our desired result, simply let $X_{n,m}:=\dfrac{X_m^{( n )} - p_n}{n p_n (1-p_n)}$.

Thanks to P.Windridge's comment, I can now answer my own question.

Indeed the convergence to standard normal in question can follow from a triangular array version of CLT called the Lindeberg-Feller CLT. Proof can be found on Durrett's Probability: Theory and Examples (freely available online).

I reference the statement of the theorem from Durrett:

To arrive at our desired result, simply let $X_{n,m}:=\dfrac{X_m^{( n )} - p_n}{\sqrt{n p_n (1-p_n)}}$.