I didn't find a printed spec for 39 fuel pumps other than output pressure is 3 to 4 psi. On some other years, Packard gave fuel pump delivery specs in addition to pressure. The delivery spec was 1 pint in 45 seconds at idle speed and that spec was consistent over the years they gave one so I would suspect it is valid for 39 too.
You could provide a temporary separate fuel supply and gravity feed the carb using a gas can or similar so the engine will run. Provide another can the fuel pump can pump into. Run the engine for the 45 seconds at idle speed and then measure the quantity of fuel delivered. If it is close to the pint then I would suspect the pump is OK.
You might also read over a thread at the main Packard club.
http://www.packardclub.org/forum/viewtopic.php?f=6&t=2123 There is a gentleman with a 40 120 with the same symptom. Several pages are in the thread and I didn't go back and read them but IIRC, he finally did find something and was able to resolve the issue. Perhaps some of his experiences will give you an idea.
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