Hello and welcome to JustAnswer.
First thing I need to tell you is that the PCM does not supply power to the fuel pump relay it supplies it with the ground that turns it on, the power is supplied from the fuse box. The PCM only controls components on the ground side, not the positive side, so first we need to determine why there is no power there. There should be 2 pins that have power, the third is the one that takes the 12 volt supply power and sends it thru the the inertia switch and then to the fuel pump, the 4th pin is the one that is grounded by the PCM. Since the engine does run when you manually power the fuel pump up I"m going to take for granted that the PCM power relay is being turned on and and the PCM is being turned on when the key is turned on.
Your vehicle uses what is called a CCRM relay box (short for constant control relay module). It contains the fuel pump relay and the WOT relay for the A/C compressor. Pin #13 uses a red wire with no tracer, it sends power to all of the relays and is powered by fuse #27 in the power distribution box and the power goes thru the PCM relay first, so if the car starts when you bypass the fuel pump, the fuse must be good or else the PCM would not come on and power up.
So check these pins and if all of them are correct, the CCRM is bad and will need to be replaced.
Pin #11, Light blue w/ orange tracer 12+ at all times
Pins #15 and #18 Black both are grounds and use the same grounding point on the vehicle.
Pin #13, red, should be 12+ in both run position and start position
Pin #5, red with a black tracer, is 12 volt + when the fuel pump is turned on.
Needless to say pin 5 will not be hot if relay is not on, but if all the other pins check out correctly as I have stated, the CCRM will need to be replaced since it cannot be serviced as a unit and they do go bad.
If standing in front of the car, the CCRM is under the black plastic cover over the radiator, just to the left of center, it is a black box with a long flat connector to it.
Any other questions, just respond here and I will be happy to assist you further.