Then I suspect a fuel delivery problem.
I replaced a fuel pump on one of the 93 2.3's that I bought and sold. A few weeks later, the guy that bought it (coworker) tells me he had to put another new fuel pump in it. Turns out, the fuel line segment that came with the Airco fuel pump that I installed, had swelled and leaked. All he knew is there was insufficient pressure/flow, so he bought another pump and then found that crappy line attached to the pump that I had installed.
Not saying that is your problem, but it does sound like your pump quits pumping. I believe on yours, the distributor sends the PIP signal to the PCM to keep the pump running.
Since you aren't afraid to run a power wire....Quick way to rule out whether the PIP signal is missing is to disconnect the inertia switch. Run a power wire to the inertia switch plug (it'll be one of the two pins) and hear the pump run constantly. Start it and see if it stays running.
Don't over think it though. It could be as simple as a bad fuel pump relay (which you'll be bypassing if you do the inertia switch wire trick).