Unfortunately there is no simple equation to calculate the chamber pressure.
One major complication is that the chamber temperature depends on the chemical reaction rate of the propellants, which is a function of the temperature, making it an iterative process. See How do you determine what the temperature will be in the combustion chamber of a rocket engine?
Sutton, 4th edition, p.181 sums up the process as follows
The combustion chamber conditions (such as chamber temperature and gas
composition) can be calculated by using the conditions of mass balance
(Equation 6-9), the pressure balance (Equation 6-10), several chemical
equilibrium conditions (Equation 6-8), and the energy balance
(Equation 6-11) and by simultaneously solving these equations (see
Reference 6-2, 6-3, or 6-4). The assumptions listed in Chapter 3 for
an ideal rocket apply here also. Complete combustion is postulated,
that is, all the propellant is reacted into suitable products, The
materials introduced as propellant are transformed adiabatically and
irreversibly to certain presumed reaction product constituents in the
amounts fixed by equilibrium coefficients, total pressure, and mass
balance at a temperature fixed by the available energy of reaction.
The unknowns in these equations are the chamber temperature T, and the
molar fractions n, of each of the f constituents in the reaction
product gases: thus the number of independent equations must equal f +
This process is fundamentally the same regardless of whether the engine is pump- or pressure-fed.