In the "Two-balloon experiment" Wikipedia article the James-Guth stress-strain relation includes an "internal (hydrostatic) pressure" $p$ (this is not the same as the air pressure $P$ in the balloon). They derive that, as the membrane is stretched, $p$ starts at some positive value and decreases as the square of the thickness.
Is $p$ an actual physical pressure that could somehow be measured inside a chunk of rubber?
I looked over the James-Guth analysis, which assumes that the rubber is incompressible and obeys a certain specific constitutive equation. This means that the principal stresses in the material are determined up to an arbitrary constant, which is typically denoted with the symbol p. As for any problem involving incompressibility of the material, the value of the "pressure" p must be determined by applying the boundary conditions. The state of stress in the rubber is considered as "plane stress," which means that the stress in the thickness direction is negligible compared to the hoop stresses in the rubber. From this, the p value is determined by setting the thickness stress equal to zero.
Here is an analysis I did a while back (post #4) of balloon expansion for a material of arbitrary elastic constitutive behavior that you may find of value: https://www.physicsforums.com/threads/hookes-law-for-a-balloon.670566/#post-4264407
Regarding the balloon inflation question, if I use the well-known Mooney-Rivlin constitutive equation for rubber for this problem (assuming incompressible), I get a relationship for the pressure difference across the balloon membrane that is very similar to the James-Guth result:
$$P=\frac{4h_0}{r_0}\frac{C_1}{\lambda}\left(1+\frac{C_2}{C_1}\lambda^2\right)\left(1-\frac{1}{\lambda^6}\right)$$where $C_1$ and $C_2$ are material constants, $h_0$ is the initial rubber thickness, $r_0$ is the initial radius, and $\lambda$ is the rubber stretch ratio $r/r_0$.