This is not a proper diagnoses.
At Idle and in park the manual valve, Assuming it is properly detented by the rooster comb, dumps almost all the line pressure into the pan. You could be running as little as 5 psi.The TC control valve would then also be seeing 5psi. And the TC could be seeing less. And the cooler is after the TC, so if the TC is not moving fluid,guess what,none comes out the line.
The proper test is to install a pressure gauge into the L/R servo and rev it up to 1000 rpm, looking for in excess of 230 psi......in REVERSE;A properly detented reverse.
I highly doubt,in this case, that you have a pump issue.
So suppose that you pull it out, and the pump is dandy fine, and the TC hub is too; then what? You now have no way to continue the diagnoses..... now you have to do a complete tear down and HOPE you find the real problem.
But suppose you don't find a problem, do you then put it back together and slam it back in,HOPING for the best? And then when it works the same, what will you do then?
I don't know about you, but I wouldn't be taking it out until I KNOW what the problem is. I hate taking trannys out in the first place, and I sure don't want to be doing it more than once.
And there is a very good chance, that this is not a mechanical issue.............
unless the L/R band is not functioning, or the manual valve is not properly detenting.
Nevertheless carry on,lol.