That is a classic example of a bad power connection. What is happening is enough current is getting to the fuses without a load and is enough power to run a test light. newer test lights do not put much of a load on the circuit. As soon as you hit the button, components suck the "surface voltage" down and your light goes out. If you leave the test light on you may notice it come on very slowly once you release the button.
As suggested the problem is probably right where the power cables come through the back. You need to try your test light on the cables coming in to the generator. I do understand you substituted the power there but I just would like to demonstrate the issue clearly. I highly suspect the light will stay nice and bright during that test.
Now, carefully remove the back panel off the generator. You must be very careful to not let it slide down onto the battery cables. Put some duct tape over them for safety. Of course you should disconnect your batteries but often when troubleshooting messing about with things cures or causes additional problems.
Now with you test light, connect the ground clip to the OUTSIDE ground lug. Carefully probe the inside connections. Do this on both the positive wire and even the negative wire. IF you get any hint of a light when testing the negative side you have a ground problem.
Assuming your ground is ok then test the positive wire. Be very careful prodding about and try your best to NOT disturb the connection. You are trying to find the problem. Fixing it comes after that.
If you get a good indication with the test probe there, have someone punch the starter button again and test. If the light stays bright, then the problem is further down the wire and probably the connection on the starter which is a common place to stack wires.
Myron & Deborah
08 HR Endeavor 40 SKQ
2012 Ford Edge Limited FWD 3.5L
Aluminator tow bar, Demco base plate