The fuel pump in that vehicle is located inside the fuel tank.
On your car though, the pump can be accessed by removing the rear seat lower cushion. There is a access plug (about 8" in diameter) located under the seat and the pump is located under there.
Once there, disconnect the fuel lines by squeezing the retainers and pulling up (gently) and then turn the locking ring. This requires either the special tool, of a hammer and screwdriver.
Once the locking ring is off, then the pump comes right out.
It is not.
It fits through there with no problems.
The lines under the top cover may be hanging up, but once they are moved the pump should pretty much fall out.
If it is not fitting through the opening, you may have to lower the fuel tank slightly to get the pump out. It all comes down to whether your vehicle is built with a returnless or non-returnless fuel system.
There is no easy way to tell which is which, without knowing what you are looking at. A returnless fuel system only has one fuel line going to the engine, a non-returnless system has two.
In that case, you are going to have to lower the fuel tank out of the car.
I cannot recommend that you cut that opening bigger.
The cougar is the one car that has a issue with the fuel pumps. It is as though it (ford) wants to trick you into thinking that the pump is easier then it is.
I would say that about half the cars have a hole big enough to fit the pump through and half don't. Once you said that the opening was 4.5", that was enough.
As a side note, the Lincoln LS has a grommet type of cover for the fuel pumps (there are 2 on the LS) that will fill in any opening that would be large enough for the pump to fit through. You might be able to get one of those covers from a wrecking yard for Cheap.
Just a thought. I know how much of a pain the tanks in those cars can be.