Good topic

As I understand from your question, you are seeking for the literature where 1-d flow models are discussed with pressure as a lumped parameter at a particular point. There are in fact some models for lumped parameter model for pressure in pipe network optimization techniques. You may refer to the following link to find some of the interesting techniques:

Conference Paper A review of pipe network optimisation techniques

There are multiple other domains where this kind of 1-D model is used. The Hemodynamics model is another example where some great 1-D models are developed for the blood flow through arteries.

The problem arises to model pressure as a lumped parameter in a full 3D incompressible flow is that not only pressure is a spatially dependent property in space but the pressure is the Lagrange multiplier which is introduced to maintain the incompressibility of the flow. That's why in the incompressible flow there is no dedicated equation for pressure. If you consider the momentum conservation relation as the equation for velocity, the solution can take the pressure as the distribution for which the velocities from the momentum conservation equation satisfies continuity. Additionally, in the 1D approximation problems, we represent velocity as volume flux (flow rate) of the fluid through the pipe and we model the skin friction coefficient as a function of that. This technique starts with an initial guess for the flow rate and that has been modified until we get the flow rate for which the skin-friction drop matches the pressure difference (or as the loss energy of the head relations in Bernoulli's equation). As we see here, this is a model that is obtained with a set of approximations and has its limitations. The lumped parameter model for pressure is much more straight forward for a compressible flow as there is a dedicated equation for pressure (equation of state). Compressible flow in a convergent-divergent nozzle is a great example of that.

In fluid dynamics analysis, there is integral approach and differential approach. The integral approach results in Bernoulli equation and its modified versions that are valid under certain assumptions. In cases where the flow needs to be resolved in spatial and temporal domains, the differential approach (Navier-Stokes equations) need to be solved. There are many correlations in open literature that can be used for certain type of fluid flow.