= Solution
Put $A=R^2\dot R$. The <partial derivative> in the <Unsteady Bernoulli equation> is taken at fixed spatial position, so
$$
\phi_t=\dot G-\frac{\dot A}{r},\qquad
|\nabla\phi|^2=\frac{A^2}{r^4}.
$$
At infinity the <velocity> vanishes and $p\to p_\infty$, giving $F(t)=\dot G+p_\infty/\rho$. Substitution cancels the arbitrary $G$ and yields the <pressure field of a pulsating sphere>:
$$
\boxed{\frac{p(r,t)}\rho
=\frac1r\frac d{dt}(R^2\dot R)
-\frac{R^4\dot R^2}{2r^4}+\frac{p_\infty}\rho}.
$$
The distinction between $\partial_t\phi$ at fixed $r$ and the <material derivative> along the moving boundary is essential here.
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