The money is obviously out of your pocket, and I encourage you to fight to get it back. I would make the dealer provide that diagnostic report. There's zero reason for you to take their word for it.
VWoA doesn't care what the dealer says when it comes to warranty claim. A service manager may have some power to argue for warranty coverage. But for a CEL the car would have to be connected to the computer for diagnostic session, online. And that report provided to get warranty coverage.
As an aside that is the reason why people with a tune, or people thinking about getting a tune, are always worried about warranty coverage. To get something covered the dealer is going to have to submit a report from that diagnostic session. And within the report will be things like the flash counter that might show that an ECU has been flashed more times than VW has record of flashing it... which would suggest a tune. None of that really matters here.
Back on topic, the only reason the dealer would not want to submit the record of that diagnostic session is if they know it would be covered by warranty. A customer is going to pay more than VWoA will pay, simple as that.
Dealer says faulty fan control module is why the fan was running all the time. If the fan is running all the time it kills the battery. So the dead battery, even if it did cause a CEL, wouldn't be the reason for the warranty claim. If the dealer doesn't want to provide a diagnostic session to VW then telling VW that a dead battery caused the CEL is specifically so that the repairs will not be covered under warranty.
There are plenty of faults and warnings you get for a dead, low, disconnected, or not charging battery. Some of them, like the not charging, will give you a frightening red triangle in the dash and tell you to stop driving immediately. But it does not cause a check engine light.
