A bit of guessing without being tested to verify?
This sounds very much like a common problem on RV. The battery is good but the path from it to the starter solenoid is dirty or corroded. That dirty connection lets enough power through to move the starter solenoid to connect the power to try to turn the starter but it doesn't have enough getting there, so appears to die!
Have you had cars that would only give you a series of clicks when trying to start? Same problem only those did not have enough power to keep the solenoid operated and it clicks every time it operates, then falls off, only to try again!
The problem with the Winnebago drawings is that they buy stripped chassis from chassis builders and then add the RV. So they look at the chassis problems as being back on the chassis builders. Those folks who build cars and trucks are very tight with their info as they want to keep their service busy!
So when we want to DIY, we have to do it the hard way without directions or drawings to show us the way.
I like to start with the battery, make sure the cables are good, clean and tight, both on the positive side as well as the ground side. Don't forget to check the ground cable where it meets the ground on the frame, etc. Often down behind the batteries?
Then move along that cable toward the starter, looking for any bulges or connections which may be dirty.
A sneaky one is a "mega fuse" as it looks like a lump if it gets covered in mud and road oil!
Mine was on the frame just inside the front wheel which is an excellent place to get dirty!
The problem with this was it let enough power through that I could see 12Volts with my meter when looking at the starter, but actually starting took much more current than my meter!
Had I taken the trouble to take the cable off the starter and measure resistance from starter end to battery end, I would have seen high resistance.
Sorry, wish I could point to where that dirty connection might be but nothing available.
IF one were lucky enough to find a manual for the specific chassis, it "might" show the path and connections??
Never gotten that far! I did it the hard way!
If the original cables with tape markers on them to ID, this chart may help to which to trace? Red marked battery to starter if the tape is still there?