Symptoms sound more like Fuel relay, AMM, or dirty AMM connector.

The Haynes* manual will not be much help, but this reponse will bump your post back up "in the running", keeping it alive. For now, I'll address what puzzled you.

1- "First problem, we turned the key to ON not running to listen for the "whirring" sound which we didn't hear."
Typical Haynes "tip". Your '88 has LH2.2 Fuel Injection. The pumps run only during cranking and running. The Key On "whirr" came with LH2.4 in the '89 240.

2- "Well, even without the fuse it kept idling for 15 minutes so we gave up"
Two problems here:
a) Fuse 6 is NOT the pump fuse for U.S. LH2.2 fuel injected cars. Only for K-jetonic (which Haynes calls "CIS") in foreign markets. The cover label is misleading.

b) There is no dedicated pump fuse, per se, only the Main FI System fuse (25 amp near the coil on your car). And when that's pulled, the engine stops instantly. So there is NO WAY to depressurize by killing the pump, as so many believe.

3- "The bigger issue now is that once we took out the fuse, the fuel pump is "whining", something it has not done before."
Can't explain that, because Fuse 6 isn't even wired to anything in your car. Did you finally open any lines and maybe induce some air?

4- "So do we need to look for a fuel pump problem?"
Possibly, but it's too early to say. Let's see if we get any other suggestions.

* In my opinion, the Bentley 240 service manual is much better organized, with far fewer errors than the Haynes.
--
Bruce Young
'93 940-NA (current), 240s (one V8), 140s, 122s, since '63.