You were sort of on the right track. First, note that differential form of the internal energy $U$ is $$dU = T \, dS - p \, dV = \left( \frac{\partial U}{\partial S} \right)_V dS + \left( \frac{\partial U}{\partial V} \right)_S dV $$ Thus, $$ p = - \left( \frac{\partial U}{\partial V} \right)_S \tag 1 $$ Now, as you suggested, let's consider $p = p(S(T,V),V)$, so that $$ \begin{eqnarray} \left( \frac{\partial p}{\partial T} \right)_V &=& \left( \frac{\partial p}{\partial S} \right)_V \left( \frac{\partial S}{\partial T} \right)_V \tag 2\ \left( \frac{\partial p}{\partial V} \right)_T &=& \left( \frac{\partial p}{\partial S} \right)_V \left( \frac{\partial S}{\partial V} \right)_T +
\left( \frac{\partial p}{\partial V} \right)_S \tag 3 \end{eqnarray} $$ Similarly, we expand the internal energy $U = U(S(T,V),V)$, so that $$ \begin{eqnarray} \left( \frac{\partial U}{\partial T} \right)_V &=& \left( \frac{\partial U}{\partial S} \right)_V \left( \frac{\partial S}{\partial T} \right)_V \tag 4 \ \left( \frac{\partial U}{\partial V} \right)_T &=& \left( \frac{\partial U}{\partial S} \right)_V \left( \frac{\partial S}{\partial V} \right)_T +
\left( \frac{\partial U}{\partial V} \right)_S \tag 5 \end{eqnarray} $$ So, beginning as you did, we rewrite (3) as $$ \left( \frac{\partial p}{\partial V} \right)_T -
\left( \frac{\partial p}{\partial V} \right)_S = \left( \frac{\partial p}{\partial S} \right)_V \left( \frac{\partial S}{\partial V} \right)_T $$ Eliminate $\partial S/\partial V$ on the right using (5) $$ \left( \frac{\partial p}{\partial V} \right)_T -
\left( \frac{\partial p}{\partial V} \right)_S = \frac{\displaystyle\left( \frac{\partial p}{\partial S} \right)_V}{\displaystyle\left( \frac{\partial U}{\partial S} \right)_V } \left[ p + \left( \frac{\partial U}{\partial V} \right)_T \right] $$ Eliminate $\partial U/\partial S$ using (4) $$ \left( \frac{\partial p}{\partial V} \right)_T -
\left( \frac{\partial p}{\partial V} \right)_S = \frac{\displaystyle\left( \frac{\partial p}{\partial S} \right)_V \left( \frac{\partial S}{\partial T} \right)_V }{\displaystyle\left( \frac{\partial U}{\partial T} \right)_V } \left[ p + \left( \frac{\partial U}{\partial V} \right)_T \right] $$ Finally, use (2) to write $$ \left( \frac{\partial p}{\partial V} \right)_T -
\left( \frac{\partial p}{\partial V} \right)_S = \frac{\displaystyle\left( \frac{\partial p}{\partial T} \right)_V}{\displaystyle\left( \frac{\partial U}{\partial T} \right)_V } \left[ p + \left( \frac{\partial U}{\partial V} \right)_T \right] $$ This is the desired form, written in a more appealing way IMO. 804