Generally, when you hold the switch in the "start" position, the control circuit is wired to apply 24v from the battery through a resistor to the exciter field to get the generator making power. Then when you release the switch, the regulated exciter current should take over. This is produced by CVT1 (large black transformer in box under output lugs) sending 3-phase AC to a rectifier board near it, making DC, which then goes to the exciter field. This current is regulated by the AC regulator board in the main control cabinet. It regulates by sending current through special windings in CVT1 that reduce CVT1's output current to the exciter. (more regulator output = less AC generator voltage out)
Something is causing this system to fail, and not supply exciter current after you let up the start switch. CVT1 could be bad, the diode board could have bad diodes, there could be broken wires or corroded connector pins, or the regulator board could have failed with its Q3 (output transistor) shorted. I'm sure I haven't thought of all the possibilities, but you should be able to use the schematic in figure 1-1 of TM5-6115-584-34.pdf to trace things through and find the problem.
The TM also has a troubleshooting section, and a paragraph that gives the resistances of all the windings of CVT1.