I understand your problem but let me explain the problem differently.
If you run two simulations, with the same downstream boundary condition, but with two different constant pressure... You expect the result change between the simulations. But this leads another question (problem?)
Normaly, the level imposed at the downstream boundary condition should also be affected by the change of the pressure.
So in fact, your expectation gives some inconsistencies between the boundary and the model...
In fact, if your domain is limited, the pressure effect will mainly drive the water level at the downstream boundary so you could take in account the pressure effect by a simple mean sea level variation estimate by the inverse barometric method
Hope this helps