Have you tried a different pump relay? Your initial post doesn't mention it but an intermittent relay could cause the pump to run then not run on initial start.
Should be R6 in the front relay box.
That car also uses the same pump control system that Ford used on many other vehicles, it varies the pump output pressure by switching the pump on/off rapidly. If it was failing it may not get the "start up" signal or not respond to it.
I would try the relay first, it's cheap and easy.
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Wednesday, November 29th, 2017 AT 6:48 PM