Shouldn't be too hard to watch the battery voltage at the battery terminals and have someone crank the GENSET. If you do see bad negative DC VOLTAGE drops with your meter touching the center of the battery terminal posts then the battery is at fault. If you don't see drastic jumps in the negative direction here then go read the voltage right at the GENSET and do the same thing again. If it jumps way down in DC VOLTAGE which is causing the clicks only then you have a bad connection between the GENSET and the battery terminals...
Keep in mind your battery will show a voltage drop but it should not be going down below 12.0VDC or so when the starter is pulling DC current. My batteries drop down real quick to around 12.3-4VDC when hit with a very large load and then jump right back up to its full charge of 12.6-7VDC when I remove the large load. When you would see DC VOLTAGE dropping way below the 12.0VDC reading is where it will start making clicky sounds like it does when your battery is dead.
The battery cables around the terminal could be at fault or any number of connection points down the line.
The GENSET will draw a whole bunch of DC CURRENT when trying to start starting and will find all of these weak connection points very quickly...
It could even be the GENSET starter motor contacts making bad connections but that probably would not show up in your DC VOLTAGE readings...
Another quick test is to nose up your truck to the GENSET and use JUMPER CABLEs from the truck start battery clipped into the GENSET DC Connection terminals and see what happens.
Once you develop an easy to understand test procedure then you can identify where things along the long leads between the battery terminals and the GENSET are going bad for you... It will go much better for you then knowing what to look at when this happens...
Just my thoughts
Roy Ken