So you have 12V at the coil terminals until you plug in the relay, then it drops? Is that 12V measured to the ground / PCM terminal of the relay or another ground?

If so, then I agree you have a problem with the ground back to or inside the PCM.

Have you checked the voltage and what it measures to the battery ground? If it does the same thing, then maybe you have a bad connection or wire on the power side. A relay coil is not much of a load, but it sounds like you have something either shorting or loosing continuity under load.

I assume you've checked for power on the contacts / fuel pump side of the relay. Does a jumper on that side run the pump and provide fuel pressure?