You have two kinds of pressure to discuss. There is dynamic pressure along the engine axis, which is what we simplistically consider when we estimate thrust. And there is also static pressure at or about the exit plane, which contributes more or less to thrust but also describes the shape of the exhaust plume with respect to ambient pressure -- in rocketry terms, behavior in a vacuum or during boost upward through an ever-thinning atmosphere.
Dynamic pressure varies across the exit plane, but you can use an average if you want and be within an order of magnitude correctness. Exhaust moves faster at the center of the exit plane. In the case of the Rocketdyne F-1, much faster. The edges of the exit plane are dominated by the film cooling regime in the nozzle extension which, by design, employs a much cooler and much slower flow. Hence if you want to talk of averages, exclude the outer 2-3 inches as they are not part of the ordinary exit-plane shear regime.
The numbers you're looking for in your estimate are 1,522,000 lbf for sea-level thrust (AS-504 and later) and 148 inches diameter (standard fitted nozzle extension). But you really also want to look at such things as mass flow rate (5683 lbm/s) and exhaust velocity (9800 f/s), because these will more accurately describe the dynamics of the exhaust hitting the ambient air. Imagine scouring away dirt from the pavement with your garden sprayer. You feel the reactive thrust against your hand, and that's measurable. But the effect of the fluid hitting the pavement in a directional flow is governed by slightly different physical regimes such as kinetic energy. If you use the numbers above to compute kinetic energy in the exhaust, you'll have a more usable number to describe the effect of it hitting ambient air. (It is no accident that the formulation for aerodynamic drag -- the force of moving fluid against an object -- is related to fluid kinetic energy.) Since the "object" in this case is also a fluid, the formulation becomes enormously more complicated (e.g., Navier-Stokes methods) if you want to work out the exact numbers.
The static pressure component doesn't matter much until you talk about thrust in a vacuum. Canonical atmospheric pressure of 14.7 psia is a measurement of the Brownian effect under those conditions, and ideally you want the exhaust plume to match that level of Brownian motion. If all the motion is linear, leading to ideal thrust in a vacuum, then the ambient atmosphere will act to constrict the plume and change its shape (overexpanded plume). The SSME had overexpanded nozzles because it was tuned for optimal thrust at higher altitude, the result being the Mach cone that forms while the engine is operating at sea level. Nearly all rocket nozzles have a fixed geometry. At altitude the F-1 nozzle underexpands the plume, leading to plume spread at high altitude. The static pressure of the plume is greater than ambient. In a vacuum, static exhaust pressure at the exit plane forms a measurable percentage of thrust, as much as half the thrust produced by the LM DPS.
