Dear Navneet,
Yes, and no. The “p”s in the final result that you state are from the independent field, but the “J” is actually the volumetric Jacobian, not the independent field!
Remember that in the 3-field formulation this term no longer comes from the linearisation of the constitutive law, but rather the linearisation of a component of the residual associated with the displacement degrees of freedom. In the end one can identify similarities in the structure of term for the 1- and 3-field formulations, but I don't think that one should try to move directly from the 1-field result to the 3-field result in the way that you wish to. I encourage you to read the books and papers that are given in the reference list for step-44. They might be able to highlight some key parts of the (somewhat tedious) derivation that we didn’t cover in sufficient detail in the tutorial documentation.
To hyper-summarise what those references will show, if one pulls everything back into the reference configuration then we end up with a displacement residual contribution that looks something like this:
R_{u} = \int \delta E : [S^{iso} + \tilde{p} J C^{-1}] dV
where \delta E is the variation of the Green-Lagrange strain tensor, S^{iso} is the isochoric part of the Piola-Kirchhoff stress tensor (derived from some constitutive law), \tilde{p} is the independent pressure response field, J = det(F) is the volumetric Jacobian, and C is the right Cauchy-Green deformation tensor. This tangent contribution that you are seeking comes from linearising the "\tilde{p} J C^{-1}” part with respect to the displacement, i.e.
\Delta_{u} R_{u} = \int \delta E : [H^{iso} + \tilde{p} \frac{\partial}{\partial E}[J C^{-1}] ] : \Delta E dV + ….
and pushing all of the tangents forward into the current configuration again.
Best,
Jean-Paul