Dimensional analysis: provide natural constants/constants with given unit possible?

[FelixSchloms]

I think the best way to explain my question is with my specific problem:
I have a dataset with a given pressure p0, temperature T0 and mass flow rate mflow and I would like to fit them to an outlet pressure.
( I know the ground truth equation, it is however computationally expensive to solve and I would like to search for an easier one)
But to get a more general solution I would like to add some known constants like specific gas constant, area of outlet and isentropic coefficient.
If I use the dimensional analysis I get some random constants with a unit to choose so it fits my output unit.
If I force the algorithm to use dimensionless constants only, it just uses p0 because combining my three input variables doesnt lead to a pressure unit.
So I was wondering if there is way to allow the algorithm to use some given constants with a given unit and additionally to use dimensionless constants that it needs to find. Because the inputs and the constants I mentioned can be combined to a pressure unit.

Did anyone try something like this or implemented a similar example?